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Thedegree of a constantis 0. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Thedegree of a polynomialis the highest degree of all its terms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Exponential Notation | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Properties of ExponentsIfa,ba,bare real numbers andm,nm,nare whole numbers, thenProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambm | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,ba,bare real numbers andm,nm,nare whole numbers, thenProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambm | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
FOIL Method for Multiplying Two BinomialsâTo multiply two binomials:Step 1.Multiply theFirstterms.Step 2.Multiply theOuterterms.Step 3.Multiply theInnerterms.Step 4.Multiply theLastterms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 1.Multiply theFirstterms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 2.Multiply theOuterterms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 3.Multiply theInnerterms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 4.Multiply theLastterms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Multiplying Two BinomialsâTo multiply binomials, use the:Distributive Property (Example 6.34)FOIL Method (Example 6.39)Vertical Method (Example 6.44) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Distributive Property (Example 6.34) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
FOIL Method (Example 6.39) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Vertical Method (Example 6.44) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Multiplying a Trinomial by a BinomialâTo multiply a trinomial by a binomial, use the:Distributive Property (Example 6.45)Vertical Method (Example 6.46) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Distributive Property (Example 6.45) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Vertical Method (Example 6.46) | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Binomial Squares PatternIfa,ba,bare real numbers,(a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2(aâb)2=a2â2ab+b2(aâb)2=a2â2ab+b2To square a binomial: square the first term, square the last term, double their product. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,ba,bare real numbers, | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
(a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2 | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
(aâb)2=a2â2ab+b2(aâb)2=a2â2ab+b2 | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To square a binomial: square the first term, square the last term, double their product. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Product of Conjugates PatternIfa,ba,bare real numbers,(aâb)(a+b)=a2âb2(aâb)(a+b)=a2âb2The product is called a difference of squares. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,ba,bare real numbers, | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
(aâb)(a+b)=a2âb2(aâb)(a+b)=a2âb2 | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
The product is called a difference of squares. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To multiply conjugates:square the first term square the last termwrite it as a difference of squares | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
square the first term square the last termwrite it as a difference of squares | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Quotient Property for Exponents:Ifaais a real number,aâ0aâ0, andm,nm,nare whole numbers, then:aman=amân,m>nandaman=1amân,n>maman=amân,m>nandaman=1amân,n>m | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifaais a real number,aâ0aâ0, andm,nm,nare whole numbers, then:aman=amân,m>nandaman=1amân,n>maman=amân,m>nandaman=1amân,n>m | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Zero ExponentIfaais a non-zero number, thena0=1a0=1. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifaais a non-zero number, thena0=1a0=1. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Quotient to a Power Property for Exponents:Ifaaandbbare real numbers,bâ0,bâ0,andmmis a counting number, then:(ab)m=ambm(ab)m=ambmTo raise a fraction to a power, raise the numerator and denominator to that power. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifaaandbbare real numbers,bâ0,bâ0,andmmis a counting number, then:(ab)m=ambm(ab)m=ambm | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To raise a fraction to a power, raise the numerator and denominator to that power. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Summary of Exponent PropertiesIfa,ba,bare real numbers andm,nm,nare whole numbers, thenProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amân,aâ0,m>naman=1anâm,aâ0,n>mZero Exponent Definitionao=1,aâ0Quotient to a Power Property(ab)m=ambm,bâ0Product PropertyamÂ... | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,ba,bare real numbers andm,nm,nare whole numbers, thenProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amân,aâ0,m>naman=1anâm,aâ0,n>mZero Exponent Definitionao=1,aâ0Quotient to a Power Property(ab)m=ambm,bâ0Product Propertyam·an=am+nPower Property(am)n=am... | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Fraction AdditionIfa,b,andca,b,andcare numbers wherecâ0câ0, thenac+bc=a+bcanda+bc=ac+bcac+bc=a+bcanda+bc=ac+bc | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,b,andca,b,andcare numbers wherecâ0câ0, thenac+bc=a+bcanda+bc=ac+bcac+bc=a+bcanda+bc=ac+bc | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Division of a Polynomial by a MonomialTo divide a polynomial by a monomial, divide each term of the polynomial by the monomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Property of Negative ExponentsIfnnis a positive integer andaâ0aâ0, then1aân=an1aân=an | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifnnis a positive integer andaâ0aâ0, then1aân=an1aân=an | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Quotient to a Negative ExponentIfa,ba,bare real numbers,bâ0bâ0andnnis an integer , then(ab)ân=(ba)n(ab)ân=(ba)n | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Ifa,ba,bare real numbers,bâ0bâ0andnnis an integer , then(ab)ân=(ba)n(ab)ân=(ba)n | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To convert a decimal to scientific notation:Step 1.Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.Step 2.Count the number of decimal places,nn, that the decimal point was moved.Step 3.Write the number as a product with a power of 10. If the original number is:greater than... | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 1.Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 2.Count the number of decimal places,nn, that the decimal point was moved. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 3.Write the number as a product with a power of 10. If the original number is:greater than 1, the power of 10 will be10n10nbetween 0 and 1, the power of 10 will be10ân10ân | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
greater than 1, the power of 10 will be10n10n | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
between 0 and 1, the power of 10 will be10ân10ân | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 4.Check. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
To convert scientific notation to decimal form:Step 1.Determine the exponent,nn, on the factor 10.Step 2.Move the decimalnnplaces, adding zeros if needed.If the exponent is positive, move the decimal pointnnplaces to the right.If the exponent is negative, move the decimal point|n||n|places to the left.Step 3.Check. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 1.Determine the exponent,nn, on the factor 10. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 2.Move the decimalnnplaces, adding zeros if needed.If the exponent is positive, move the decimal pointnnplaces to the right.If the exponent is negative, move the decimal point|n||n|places to the left. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
If the exponent is positive, move the decimal pointnnplaces to the right. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
If the exponent is negative, move the decimal point|n||n|places to the left. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
Step 3.Check. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-concepts |
binomial : A binomial is a polynomial with exactly two terms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
conjugate pair : A conjugate pair is two binomials of the form(aâb),(a+b)(aâb),(a+b); the pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
degree of a constant : The degree of any constant is 0. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
degree of a polynomial : The degree of a polynomial is the highest degree of all its terms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
degree of a term : The degree of a term is the exponent of its variable. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
monomial : A monomial is a term of the formaxmaxm, whereaais a constant andmmis a whole number; a monomial has exactly one term. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
negative exponent : Ifnnis a positive integer andaâ0aâ0, thenaân=1anaân=1an. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
polynomial : A polynomial is a monomial, or two or more monomials combined by addition or subtraction. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
scientific notation : A number is expressed in scientific notation when it is of the formaÃ10naÃ10nwhereaâ¥1anda<10aâ¥1anda<10andnnis an integer. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
standard form : A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
trinomial : A trinomial is a polynomial with exactly three terms. | https://openstax.org/books/elementary-algebra-2e/pages/6-key-terms |
Finding the Greatest Common Factor (GCF):To find the GCF of two expressions:Step 1.Factor each coefficient into primes. Write all variables with exponents in expanded form.Step 2.List all factorsâmatching common factors in a column. In each column, circle the common factors.Step 3.Bring down the common factors that a... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Factor each coefficient into primes. Write all variables with exponents in expanded form. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.List all factorsâmatching common factors in a column. In each column, circle the common factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Bring down the common factors that all expressions share. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Multiply the factors as inExample 7.2. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor the Greatest Common Factor from a Polynomial:To factor a greatest common factor from a polynomial:Step 1.Find the GCF of all the terms of the polynomial.Step 2.Rewrite each term as a product using the GCF.Step 3.Use the âreverseâ Distributive Property to factor the expression.Step 4.Check by multiplying the ... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Find the GCF of all the terms of the polynomial. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Rewrite each term as a product using the GCF. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Use the âreverseâ Distributive Property to factor the expression. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Check by multiplying the factors as inExample 7.5. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor by Grouping:To factor a polynomial with 4 four or more termsStep 1.Group terms with common factors.Step 2.Factor out the common factor in each group.Step 3.Factor the common factor from the expression.Step 4.Check by multiplying the factors as inExample 7.15. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Group terms with common factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Factor out the common factor in each group. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Factor the common factor from the expression. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Check by multiplying the factors as inExample 7.15. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor trinomials of the formx2+bx+cx2+bx+cStep 1.Write the factors as two binomials with first termsx:(x)(x)(x)(x).Step 2.Find two numbersmandnthatMultiply toc,m·n=cm·n=cAdd tob,m+n=bm+n=bStep 3.Usemandnas the last terms of the factors:(x+m)(x+n)(x+m)(x+n).Step 4.Check by multiplying the factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Write the factors as two binomials with first termsx:(x)(x)(x)(x). | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Find two numbersmandnthatMultiply toc,m·n=cm·n=cAdd tob,m+n=bm+n=b | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Usemandnas the last terms of the factors:(x+m)(x+n)(x+m)(x+n). | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Check by multiplying the factors. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor Trinomials of the Formax2+bx+cax2+bx+cusing Trial and Error:SeeExample 7.33.Step 1.Write the trinomial in descending order of degrees.Step 2.Find all the factor pairs of the first term.Step 3.Find all the factor pairs of the third term.Step 4.Test all the possible combinations of the factors until the correct pr... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Write the trinomial in descending order of degrees. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Find all the factor pairs of the first term. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Find all the factor pairs of the third term. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Test all the possible combinations of the factors until the correct product is found. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 5.Check by multiplying. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Factor Trinomials of the Formax2+bx+cax2+bx+cUsing the âacâ Method:SeeExample 7.38.Step 1.Factor any GCF.Step 2.Find the product ac.Step 3.Find two numbersmandnthat:Multiply toacm·n=a·cAdd tobm+n=bMultiply toacm·n=a·cAdd tobm+n=bStep 4.Split the middle term usingmandn:Step 5.Factor by grouping.Step 6.Check by m... | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 1.Factor any GCF. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 2.Find the product ac. | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 3.Find two numbersmandnthat:Multiply toacm·n=a·cAdd tobm+n=bMultiply toacm·n=a·cAdd tobm+n=b | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
Step 4.Split the middle term usingmandn: | https://openstax.org/books/elementary-algebra-2e/pages/7-key-concepts |
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