39 Practice: Operations with Polynomials (6.4 – 6.7) — Intermediate Algebra

Answer keys located in next section.

6.4 Basic Operations Using Polynomials

For questions 1 to 8, simplify each expression using the variables given.

$-a^3 - a^2 + 6a - 21 \text{ when }a = -4$
$n^2 + 3n - 11 \text{ when }n = -6$
$-5n^4 - 11n^3 - 9n^2 - n - 5 \text{ when } n = -1$
$x^4 - 5x^3 - x + 13 \text{ when } x = 5$
$x^2 + 9x + 23 \text{ when } x = -3$
$-6x^3 + 41x^2 - 32x + 11 \text{ when } x = 6$
$x^4 - 6x^3 + x^2 - 24 \text{ when } x = 6$
$m^4 + 8m^3 + 14m^2 + 13m + 5 \text{ when } m = -6$

For questions 9 to 28, simplify the following expressions.

$(5p - 5p^4) - (8p - 8p^4)$
$(7m^2 + 5m^3) - (6m^3 - 5m^2)$
$(1 + 5p^3) - (1 - 8p^3)$
$(6x^3 + 5x) - (8x + 6x^3)$
$(5n^4 + 6n^3) + (8 - 3n^3 - 5n^4)$
$(8x^2 + 1) - (6 - x^2 - x^4)$
$(2a + 2a^4) - (3a^2 - 5a^4 + 4a)$
$(6v + 8v^3) + (3 + 4v^3 - 3v)$
$(4p^2 - 3 - 2p) - (3p^2 - 6p + 3)$
$(7 + 4m + 8m^4) - (5m^4 + 1 + 6m)$
$(3 + 2n^2 + 4n^4) + (n^3 - 7n^2 - 4n^4)$
$(7x^2 + 2x^4 + 7x^3) + (6x^3 - 8x^4 - 7x^2)$
$(8r^4 - 5r^3 + 5r^2) + (2r^2 + 2r^3 - 7r^4 + 1)$
$(4x^3 + x - 7x^2) + (x^2 - 8 + 2x + 6x^3)$
$(2n^2 + 7n^4 - 2) + (2 + 2n^3 + 4n^2 + 2n^4)$
$(7b^3 - 4b + 4b^4) - (8b^3 - 4b^2 + 2b^4 - 8b)$
$(8 - b + 7b^3) - (3b^4 + 7b - 8 + 7b^2) + (3 - 3b + 6b^3)$
$(1 - 3n^4 - 8n^3) + (7n^4 + 2 - 6n^2 + 3n^3) + (4n^3 + 8n^4 + 7)$
$(8x^4 + 2x^3 + 2x) + (2x + 2 - 2x^3 - x^4) - (x^3 + 5x^4 + 8x)$
$(6x - 5x^4 - 4x^2) - (2x - 7x^2 - 4x^4 - 8) - (8 - 6x^2 - 4x^4)$

6.5 Multiplication of Polynomials

Find each product.

$6(p - 7)$
$4k(8k + 4)$
$2(6x + 3)$
$3n^2(6n + 7)$
$(4n + 6)(8n + 8)$
$(2x + 1)(x - 4)$
$(8b + 3)(7b - 5)$
$(r + 8)(4r + 8)$
$(3v - 4)(5v - 2)$
$(6a + 4)(a - 8)$
$(5x + y)(6x - 4y)$
$(2u + 3v)(8u - 7v)$
$(7x + 5y)(8x + 3y)$
$(5a + 8b)(a - 3b)$
$(r - 7)(6r^2 - r + 5)$
$(4x + 8)(4x^2 + 3x + 5)$
$(6n - 4)(2n^2 - 2n + 5)$
$(2b - 3)(4b^2 + 4b + 4)$
$(6x + 3y)(6x^2 - 7xy + 4y^2)$
$(3m - 2n)(7m^2 + 6mn + 4n^2)$
$(8n^2 + 4n + 6)(6n^2 - 5n + 6)$
$(2a^2 + 6a + 3)(7a^2 - 6a + 1)$
$(5k^2 + 3k + 3)(3k^2 + 3k + 6)$
$(7u^2 + 8uv - 6v^2)(6u^2 + 4uv + 3v^2)$
$(2n^3 - 8n^2 + 3n + 6)(n^3 - 6n^2 - 2n + 3)$
$(a^3 + 2a^2 + 3a + 3)(a^3 + 2a^2 - 4a + 1)$
$3(3x - 4)(2x + 1)$
$5(x - 4)(2x - 3)$
$3(2x + 1)(4x - 5)$
$2(4x + 1)(2x - 6)$

6.6 Special Products

Find each product.

$(x + 8)(x - 8)$
$(a - 4)(a + 4)$
$(1 + 3p)(1 - 3p)$
$(x - 3)(x + 3)$
$(1 - 7n)(1 + 7n)$
$(8m + 5)(8m - 5)$
$(4y - x)(4y + x)$
$(7a + 7b)(7a - 7b)$
$(4m - 8n)(4m + 8n)$
$(3y - 3x)(3y + 3x)$
$(6x - 2y)(6x + 2y)$
$(1 + 5n)^2$
$(a + 5)^2$
$(x - 8)^2$
$(1 - 6n)^2$
$(4x - 5)^2$
$(5m - 8)^2$
$(3a + 3b)^2$
$(5x + 7y)^2$
$(4m - n)^2$
$(5 + 2r)^2$
$(m - 7)^2$
$(4v - 7)(4v + 7)$
$(b + 4)(b - 4)$

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