Question

simplifying polynomials
directions: simplify each polynomial. write all answers in standard form.
10 ( (2x^3 + 3x + 2) - (x^3 - 4x - 1) )
11 ( 2a^3(5a^2 - 7a) + (3a^4 - a^5) )
12 ( 5c^3d(9c^3d^2 - 4c^3d - 2) )
13 ( -4y(5y - 11) - 2y(3y + 10) )
14 ( (x - 1)(4x - 3) )
15 ( (5y - 1)^2 )
16 ( (2x + 5y)(x - 3y) )
17 ( (x + 4)(4x - 2) - 2x^2 )
18 ( \frac{-8x^5 + 28x^4 - 4x^3}{4x^3} )
19 ( \frac{27c^4d^7 + 15c^3d^5}{3c^3d^5} )

Explanation:

Step1: Distribute the negative sign

$2x^2 + 3x + 2 - x^2 + 4x + 1$

Step2: Combine like terms

$(2x^2 - x^2) + (3x + 4x) + (2 + 1)$
$x^2 + 7x + 3$

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Step1: Distribute the monomial

$2a^3 \cdot 5a^2 - 2a^3 \cdot 7a + 3a^4 - a^5$
$10a^5 - 14a^4 + 3a^4 - a^5$

Step2: Combine like terms

$(10a^5 - a^5) + (-14a^4 + 3a^4)$
$9a^5 - 11a^4$

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Step1: Distribute the monomial

$5c^2d \cdot 9c^2d^2 - 5c^2d \cdot 4c^3d - 5c^2d \cdot 2$
$45c^4d^3 - 20c^5d^2 - 10c^2d$

Step2: Rearrange to standard form

$-20c^5d^2 + 45c^4d^3 - 10c^2d$

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Step1: Distribute both monomials

$-4y \cdot 5y + (-4y) \cdot (-11) - 2y \cdot 3y - 2y \cdot 10$
$-20y^2 + 44y - 6y^2 - 20y$

Step2: Combine like terms

$(-20y^2 - 6y^2) + (44y - 20y)$
$-26y^2 + 24y$

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Step1: Use FOIL method

$x \cdot 4x + x \cdot (-3) -1 \cdot 4x + (-1) \cdot (-3)$
$4x^2 - 3x - 4x + 3$

Step2: Combine like terms

$4x^2 - 7x + 3$

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Step1: Expand the square

$(5y - 1)(5y - 1)$
$5y \cdot 5y + 5y \cdot (-1) -1 \cdot 5y + (-1) \cdot (-1)$
$25y^2 - 5y - 5y + 1$

Step2: Combine like terms

$25y^2 - 10y + 1$

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Step1: Use FOIL method

$2x \cdot x + 2x \cdot (-3y) + 5y \cdot x + 5y \cdot (-3y)$
$2x^2 - 6xy + 5xy - 15y^2$

Step2: Combine like terms

$2x^2 - xy - 15y^2$

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Step1: Use FOIL method first

$x \cdot 4x + x \cdot (-2) + 4 \cdot 4x + 4 \cdot (-2) - 2x^2$
$4x^2 - 2x + 16x - 8 - 2x^2$

Step2: Combine like terms

$(4x^2 - 2x^2) + (-2x + 16x) - 8$
$2x^2 + 14x - 8$

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Step1: Split the fraction

$\frac{-8x^5}{4x^3} + \frac{28x^4}{4x^3} - \frac{4x^3}{4x^3}$

Step2: Simplify each term

$-2x^2 + 7x - 1$

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Step1: Split the fraction

$\frac{27c^4d^7}{3c^2d^5} + \frac{15c^3d^5}{3c^2d^5}$

Step2: Simplify each term

$9c^2d^2 + 5c$

Answer:

1. $x^2 + 7x + 3$
2. $9a^5 - 11a^4$
3. $-20c^5d^2 + 45c^4d^3 - 10c^2d$
4. $-26y^2 + 24y$
5. $4x^2 - 7x + 3$
6. $25y^2 - 10y + 1$
7. $2x^2 - xy - 15y^2$
8. $2x^2 + 14x - 8$
9. $-2x^2 + 7x - 1$
10. $9c^2d^2 + 5c$