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1.(a + 7)(2a2 + 5a + 3)

2.(2b + 3)(3b2 + 2b + 5)

3.(c − 8)(4c2 − 7c − 2)

4.(2x − 1)(4x3 − 7x2 + 5x − 7)

5.(y + 4)(y2 − 5y + 1)

6.(x − 2)(x2 − 4x + 4)

7.(t − 2)(t2 + 2t + 4)

8.(x + 1)(x2 − x + 1)

9.(x2 − 4)(x2 + 4x + 4)

10.(y2 − 3y + 5)(2y2 + 5y − 3)

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Dividing polynomials

Polynomial division is built on dividing monomials, but there are systems to help organize larger problems.

Dividing by a monomial

To divide a monomial by a monomial, divide the coefficients and use the rules for exponents to simplify like variables.

To divide a larger polynomial by a monomial, divide each term of the larger polynomial by the monomial divisor.

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EXERCISE 6.9

Divide the polynomials and give your answer in simplest form.

1.(24c7d2) ÷(3c4)

2.(−35d6) ÷(−7d2)

3.(−52x9) ÷(−13x2)

4.(20y12) ÷(4y8)

5.(24x5y) ÷(−6x3)

6.(15x2) ÷(15x2)

7.(−3.9t6) ÷(1.3t4)

8.(−18q7r3) ÷(q7r)

9.(8x4 − 9x7) ÷(x3)

10.(15y5 − 21y7) ÷(3y2)

11.(5x2 + 15x) ÷(5x)

12.(−9y3 + 13y4) ÷(−y3)

13.(56z10 − 49z9 + 42z8 − 35z7) ÷(7z2)

14.(−9x4y3 + 27x3y4 − 81x2y5) ÷(−3xy)

15.(15x3 − 5x2 + 20x) ÷(5x)

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Long division

Long division of polynomials is modeled on the algorithm for long division that you learned in arithmetic. It can be used to divide by a monomial, but it is more commonly used when the divisor is a binomial or a larger polynomial.

Arrange the dividend and the divisor in standard form, highest power to lowest, and insert 0s for any missing powers to make it easier to line up like terms. Divide the first term of the dividend by the first term of the divisor and place the result as the first term of the quotient. Multiply the entire divisor by the term you just placed in the quotient, aligning like terms under the dividend. Subtract, and bring down any remaining terms in the dividend.

Repeat those steps, but use this new expression formed by subtracting and bringing down as your dividend.

You can express the remainder as a fraction by putting the remainder as the numerator of the fraction and the divisor as the denominator. The division problem(6x3 + 8x + 10)÷(2x + 4) is equal to .

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EXERCISE 6.10

Divide using long division.

1.(x2 − 15x + 56) ÷(x − 8)

2.(y2 − y − 20) ÷(y − 5)

3.(6x2 + 5x − 6) ÷(3x − 2)

4.(84x4 − 3x2 − 45) ÷(12x2 − 9)

5.(9x2 − 42x + 45) ÷(3x − 8)

6.(2a2 + 7a + 5) ÷(a + 3)

7.(2b2 − 7b + 3) ÷(b − 3)

8.(12x3 + 17x2 − 20x − 20) ÷(3x + 5)

9.(x4 + 8x2 + 12) ÷(x2 + 2)

10.(8y3 + 1) ÷(2y + 1)

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•7•

Factoring

Factoring is the process of reexpressing a quantity as the product of two or more quantities, called factors. You can factor 35 by writing it as 5 × 7, and you can factor a monomial like −3x2 y3 by writing out −3 · x · x · y · y · y. Factoring polynomials is a little more complicated, but a few rules will cover most situations.

Greatest common monomial factor

You will often be able to find several different factor pairs for a monomial. You know this is true when you factor a constant. The constant 24 could be factored as 1 × 24, 2 × 12, 3 × 8, or 4 × 6. It’s also true for monomials that involve variables. The monomial 48x5 could be written as 48 · x5, 12x2 · 4x3, 3x · 16x4, and more.

The greatest common monomial factor of a polynomial is the largest monomial that is a factor of every term. In this context, largest means the “largest coefficient and the highest power of the variable.” The polynomial 12x5y + 8x4y2 − 10x3y3 has a greatest common factor of 2x3y because 2 is the largest integer that divides all three coefficients, x3 is the largest power of x present in all terms, and y is the largest power of y present in all terms. Notice that the largest power of a variable contained in all the terms is the smallest one you see. The largest power of x in 12x5y + 8x4y2 − 10x3y3 is x3, which is the smallest power of x in any of the terms.

Factoring out the greatest common factor is applying the distributive property in reverse. Bring the common factor