1. Fundamental Concepts of Algebra

Perform Operations with Polynomials in Several Variables

Polynomials

Polynomials: Adding, Subtracting, Multiplying and Simplifying - Example 1

Perform the indicated operations. See Examples 2–6. (3p+5)^2

The special products can be used to perform selected multiplications. On the left, we use (x+y)(x-y) = x^2-y^2. On the right, (x-y)^2 = x^2-2xy+y^2. Use special products to evaluate each expression. < SEE SAMPLE B> 71^2

In Exercises 83–90, perform the indicated operation or operations. (3x+4y)^2−(3x−4y)^2

In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. A number decreased by the sum of the number and four

In Exercises 15–58, find each product. (x−1)(x+2)

Fill in the blank to correctly complete each sentence. In the term -6x^2y, -6 is the ______.

Find each product. See Examples 5 and 6. (z-3)^3

Find each product. See Examples 3–5. (x+1)(x+1)(x-1)(x-1)

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (y^2+7y−18)/(y^2−3y+2)

In Exercises 15–32, multiply or divide as indicated. (x+5)/7 ÷ (4x+20)/9

Find each product. See Examples 3–5. (3w+2)(-w^2+4w-3)

In Exercises 15–58, find each product. (x+5)(x−5)

Find each product. See Examples 5 and 6. (4x^2-5y0(4x^2+5y)

Polynomials: Adding, Subtracting, Multiplying and Simplifying - Example 3

Find each product. See Examples 5 and 6. (r+3)^4

Multiplying Polynomials - Slightly Harder Examples #6

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. -5x^11

In Exercises 83–90, perform the indicated operation or operations. (2x−7)^5/(2x−7)^3

In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. Eight decreased by three times the sum of a number and six

Adding and Subtracting Polynomials

Find each product. See Examples 5 and 6. [(3q+5)-p][(3q+5)+p]

Find each product. See Examples 3–5. (5m-6)(3m+4)

Find each product. See Examples 5 and 6. [(9r-s)+2][(9r-s)-2]

Find each product. See Examples 3–5. (m-n+k)(m+2n-3k)

Use FOIL Method to Multiply Polynomials Algebra

In Exercises 33–68, add or subtract as indicated. (3x+2)/(3x+4) + (3x+6)/(3x+4)

Multiply polynomials using distributive property

In Exercises 67–82, find each product. (3x−y)(2x+5y)

In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. (x−1)/(x^2+11x+10)

Find each product. See Examples 5 and 6. (q-2)^4

In Exercises 5–8, find the degree of the polynomial. 3x^2−5x+4

In Exercises 15–58, find each product. (7x^3+5)(x^2−2)

Find each product. See Examples 3–5. (3y-5)(3y+5)(9y^2-25)

Find each product. See Examples 5 and 6. (y+2)^3

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (−6x^3+5x^2−8x+9)+(17x^3+2x^2−4x−13)

Perform Operations with Polynomials of Several Variables

Multiplying Polynomials - Slightly Harder Examples  #1

In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. Six times the product of negative five and a number

In Exercises 15–32, multiply or divide as indicated. (x^2+x)/(x^2−4) ÷ (x^2−1)/(x^2+5x+6)

Introduction to Polynomials

In Exercises 121–128, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between the product of six and a number and negative two times the number

Find each product. See Examples 5 and 6. [(2m+7)-n]^2

Subtract polynomials by combining like terms and using integer operations

Multiplying binomials in two variables

Multiply polynomials using a box

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. 5

Polynomial... or NOT?! Recognizing Polynomials, the degree and some Terminology

In Exercises 15–32, multiply or divide as indicated. (x^2−4)/(x−2) ÷ (x+2)/(4x−8)

Perform the indicated operations. See Examples 2–6. (7m+2n)^2

In Exercises 15–58, find each product. (5−7x)(5+7x)

Add or subtract, as indicated. See Example 2. (5x^2-4x+7) + (-4x^2+3x-5)

Find each product. See Examples 3–5. 4x^2(3x63+2x^2-5x+1)

Multiplying Binomials by the FOIL Method

How to Multiply Binomials

Find each product. See Examples 5 and 6. (a-6b)^2

Find each product. See Examples 5 and 6. (2m+3)(2m-3)

Find each product. See Examples 5 and 6. (2z^4-3y)^2

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (-6x^3+7x^2-9x+3)+(14x^3+3x^2-11x-7)

Expanding Square Brackets

In Exercises 15–58, find each product. (x+5)^2

Multiplying Polynomials

Perform the indicated operations. See Examples 2–6. (x^4-3x^2+2) - (-2x^4+x^2-3)

The special products can be used to perform selected multiplications. On the left, we use (x+y)(x-y) = x^2-y^2. On the right, (x-y)^2 = x^2-2xy+y^2. Use special products to evaluate each expression. <SEE SAMPLE A> x = 63 y = 57

Find each product. See Examples 3–5. (2z-1)(-z^2+3z-4)

Perform the indicated operations. Indicate the degree of the resulting polynomial. (13x^3y^2-5x^2y-9x^2)-(-11x^3y^2-6x^2y+3x^2-4)

Find each product. See Examples 5 and 6. (8s-3t)(8s+3t)

In Exercises 15–58, find each product. (x−3)^2

Find each product. See Examples 3–5. (r-3s+t)(2r-s+t)

In Exercises 83–90, perform the indicated operation or operations. (3x+5)(2x−9)−(7x−2)(x−1)

Find each product. Assume all variables represent positive real numbers. (p^1/2-p^-1/2)(p^1/2+p^-1/2)

MATHS - Algebra - Polynomials

Polynomials: Adding, Subtracting, Multiplying and Simplifying - Example 2

Multiplying Binomials

In Exercises 5–8, find the degree of the polynomial. x^2−4x^3+9x−12x^4+63

In Exercises 67–82, find each product. (9x+7y)^2

In Exercises 33–68, add or subtract as indicated. (x+9)/10x^3 + 11/15x^2

Classify polynomials

In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (7x^4 y^2−5x^2 y^2+3xy)+(−18x^4 y^2−6x^2 y^2−xy)

Perform the indicated operations. See Examples 2–6. -z^3(9-z) + 4z(2+3z)

Find each product. See Examples 5 and 6. (5r-3t^2)^2

In Exercises 15–58, find each product. (9−5x)^2

In Exercises 33–68, add or subtract as indicated. 3/(x+1) − 3/x

How to Multiply Polynomials Properly - Nerdstudy

Add or subtract, as indicated. See Example 2. -(8x^3+x-3) + (2x^3+x^2) - (4x^2+3x-1)

Perform the indicated operations. See Examples 2–6. m(5m-2) + 9(5-m)

Find each product. See Examples 5 and 6. (4m+2n)^2

Find each product. Assume all variables represent positive real numbers. -4k(k^7/3-6k^1/3)

In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. (x+5)/(x^2−25)

In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. 2x+3x^2−5

Perform the indicated operations. Indicate the degree of the resulting polynomial. (7x^2-8xy+y^2)+(-8x^2-9xy-4y^2)

Add polynomials by combining like terms

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. 15a^2b^3+12a^3b^8-13b^5+12b^6

Find each product. See Examples 5 and 6. [(3a+b)-1]^2

Multiplying Multivariable Polynomials

Find each product. See Examples 3–5. (2x+3)(2x-3)(4x^2-9)

Find each product. Assume all variables represent positive real numbers. (x+x^1/2)(x-x^1/2)

Add or subtract, as indicated. See Example 2. (3m^5-3m^2+4) + (-2m^3-m^2+6)

The special products can be used to perform selected multiplications. On the left, we use (x+y)(x-y) = x^2-y^2. On the right, (x-y)^2 = x^2-2xy+y^2. Use special products to evaluate each expression. <SEE SAMPLE B> 102^2

In Exercises 67–82, find each product. (7xy^2−10y)(7xy^2+10y)

In Exercises 59–66, perform the indicated operations. Indicate the degree of the resulting polynomial. (3x^4 y^2+5x^3 y−3y)−(2x^4 y^2−3x^3 y−4y+6x)

In Exercises 15–58, find each product. (x+1)(x^2−x+1)

In Exercises 15–58, find each product. (x+2)^3

Multiplying Polynomials - Slightly Harder Examples # 5 - Cubing Binomials

Prove the Algebraic Squares Proof algebraically

Add or subtract, as indicated. See Example 2. 3(8p^2-5p) - 5(3p^2-2p+4)

In Exercises 67–82, find each product. (7x^2 y+1)(2x^2 y−3)

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. 6x+3x^4

In Exercises 15–32, multiply or divide as indicated. (x^3−25x)/4x^2 ⋅ (2x^2−2)/(x^2−6x+5) ÷ (x^2+5x)/(7x+7)

In Exercises 33–68, add or subtract as indicated. 5/x + 3

In Exercises 15–58, find each product. (1−y^5)(1+y^5)

In Exercises 33–68, add or subtract as indicated. 3x/8 + x/12

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (5x^2−7x−8)+(2x^2−3x+7)−(x^2−4x−3)

In Exercises 15–58, find each product. (x−1)^3

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. (3/8)x^5-(1/x^2)+9

In Exercises 15–32, multiply or divide as indicated. (x^2−4)/(x^2+3x−10) ÷ (x^2+5x+6)/(x^2+8x+15)

Multiplying Polynomials - Math Tutorial

Evaluate each algebraic expression for the given value or value(s) of the variable(s). 3+6(x-2)^3 for x=4

Find each product. See Examples 5 and 6. [(2p-3)+q]^2

Find each product. Assume all variables represent positive real numbers. y^5/8(y^3/8-10y^11/8)

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. (2/3)t^6+(3/t^5)+1

Find each product. See Examples 3–5. x^2(3x-2)(5x+1)

In Exercises 15–58, find each product. (4x^2−1)^2

Perform the indicated operations. See Examples 2–6. 2(3r^2+4r+2) - 3(-r^2+4r-5)

In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. (2x+3)/x

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x^2−14x+49)/(x^2−49)

Multiplying Polynomials Example Problem

Find each product. See Examples 3–5. 2b^3(b^2-4b+3)

In Exercises 15–58, find each product. (2x−3)(x^2−3x+5)

In Exercises 33–68, add or subtract as indicated. (4x^2+x−6)/(x^2+3x+2)−3x/(x+1)+5/(x+2)

Perform the indicated operations. (10m^4-4m^2)/2m

Multiplying Polynomials - Slightly Harder Examples # 4 - Cubing Binomials

Special Polynomial Products

In Exercises 15–32, multiply or divide as indicated. (x^3−8)/(x^2−4) ⋅ (x+2)/3x

In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression. 7/(x−3)

In Exercises 15–58, find each product. (4x^2+5x)(4x^2−5x)

In Exercises 67–82, find each product. (x−y)(x^2+xy+y^2)

In Exercises 15–58, find each product. (2x−5)(7x+2)

In Exercises 33–68, add or subtract as indicated. 2/5x − (x+1)/4x

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (3x−9)/(x^2−6x+9)

In Exercises 15–32, multiply or divide as indicated. (x^2−4)/(x^2−4x+4) ⋅ (2x−4)/(x+2)

Perform the indicated operations. See Examples 2–6. -3(4q^2-3q+2) + 2(-q^2+q-4)

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (17x^3−5x^2+4x−3)−(5x^3−9x^2−8x+11)

Introductory Algebra - Polynomials in Several Variables

In Exercises 15–32, multiply or divide as indicated. (6x+9)/(3x−15) ⋅ (x−5)/(4x+6)

In Exercises 33–68, add or subtract as indicated. 3x/(x−3) − (x+4)/(x+2)

In Exercises 15–58, find each product. (x+7)(x+3)

In Exercises 15–58, find each product. (7x^2−2)(3x^2−5)

Perform the indicated operations. (2x^2-x)+(x^2+4x)

Find each product. See Examples 3–5. (14r-1)(17r+2)

Perform the indicated operations. -2x^3(x^4-8)

In Exercises 33–68, add or subtract as indicated. (4x−10)/(x−2) − (x−4)/(x−2)

Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these.See Example 1. -7z^5-2z^3+1

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