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Math  /  Algebra

Questionexpression to a polynomial in standard form: (3x+2)(−x2−3x+1)(3 x+2)\left(-x^{2}-3 x+1\right)

Studdy Solution

STEP 1

1. We are given a product of two expressions and need to expand it into a polynomial.
2. The goal is to express the polynomial in standard form, which means arranging terms in descending order of powers.

STEP 2

1. Distribute each term in the first expression to each term in the second expression.
2. Combine like terms to simplify the polynomial.
3. Arrange the polynomial in standard form.

STEP 3

Distribute each term in (3x+2) (3x + 2) to each term in (−x2−3x+1) (-x^2 - 3x + 1) .
(3x+2)(−x2−3x+1)=3x(−x2)+3x(−3x)+3x(1)+2(−x2)+2(−3x)+2(1) (3x + 2)(-x^2 - 3x + 1) = 3x(-x^2) + 3x(-3x) + 3x(1) + 2(-x^2) + 2(-3x) + 2(1)

STEP 4

Calculate each of the distributed terms:
3x(−x2)=−3x3 3x(-x^2) = -3x^3 3x(−3x)=−9x2 3x(-3x) = -9x^2 3x(1)=3x 3x(1) = 3x 2(−x2)=−2x2 2(-x^2) = -2x^2 2(−3x)=−6x 2(-3x) = -6x 2(1)=2 2(1) = 2
Combine these results:
−3x3−9x2+3x−2x2−6x+2 -3x^3 - 9x^2 + 3x - 2x^2 - 6x + 2

STEP 5

Combine like terms:
−3x3+(−9x2−2x2)+(3x−6x)+2 -3x^3 + (-9x^2 - 2x^2) + (3x - 6x) + 2
This simplifies to:
−3x3−11x2−3x+2 -3x^3 - 11x^2 - 3x + 2

STEP 6

The polynomial is now in standard form:
−3x3−11x2−3x+2 \boxed{-3x^3 - 11x^2 - 3x + 2}

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