Q. \( (-x+2)(x^2+9x-2) \)

Answer

We expand by distributing:

\[
(-x+2)(x^2+9x-2)
= -x\cdot x^2 – x\cdot 9x – x\cdot(-2) + 2\cdot x^2 + 2\cdot 9x + 2\cdot(-2)
\]
\[
= -x^3 -9x^2 +2x +2x^2 +18x -4
= -x^3 -7x^2 +20x -4.
\]

Final result: \(\boxed{-x^3-7x^2+20x-4}\).

Detailed Explanation

Problem

\( (-x+2)(x^2+9x-2) \)

Step-by-step solution

Use the distributive property to split the product:

\[
(-x+2)(x^2+9x-2)=(-x)(x^2+9x-2)+2(x^2+9x-2).
\]
Multiply each term in the parentheses by \(-x\):

\[
(-x)(x^2)=-x^3,\qquad (-x)(9x)=-9x^2,\qquad (-x)(-2)=2x.
\]
Multiply each term in the parentheses by \(2\):

\[
2(x^2)=2x^2,\qquad 2(9x)=18x,\qquad 2(-2)=-4.
\]
Combine all terms:

\[
-x^3-9x^2+2x+2x^2+18x-4.
\]
Collect like terms:

\[
-x^3+(-9x^2+2x^2)+(2x+18x)-4=-x^3-7x^2+20x-4.
\]

Final answer: \(\displaystyle -x^3-7x^2+20x-4\)

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FAQs

How do I expand \( (-x+2)(x^2+9x-2) \)?

-A: Use the distributive property: \((-x)(x^2+9x-2)+2(x^2+9x-2)=-x^3-9x^2+2x+2x^2+18x-4\). Combine like terms to get \(-x^3-7x^2+20x-4\).

Can I use FOIL here?

-A: FOIL is for two binomials. For a binomial times a trinomial, use distributive law (multiply each term of the binomial by every term of the trinomial).

How do I combine like terms correctly?

-A: Collect same-power terms: from expansion \(-x^3-9x^2+2x+2x^2+18x-4\), combine \( -9x^2+2x^2=-7x^2\) and \(2x+18x=20x\).

How can I factor the expanded result back?

-A: \(-x^3-7x^2+20x-4 = -(x^3+7x^2-20x+4)\). Using root \(x=2\) gives \((x-2)(x^2+9x-2)\), so original is \((-x+2)(x^2+9x-2)\).

What are the roots of the product?

-A: Solve \((-x+2)(x^2+9x-2)=0\). Roots: \(x=2\) and \(x=\frac{-9\pm\sqrt{89}}{2}\) (from \(x^2+9x-2=0\)).

What is the degree and leading coefficient?

-A: The expanded polynomial \(-x^3-7x^2+20x-4\) has degree 3 and leading coefficient \(-1\).

How do I check my expansion quickly?

-A: Substitute an easy value, e.g. \(x=0\): original gives \(2(-2)=-4\); expanded gives \(-4\). Matching values increase confidence.

Is \(-x+2\) the same as \(2-x\) or \(-(x-2)\)?

-A: Yes: \(-x+2=2-x\), and \(-(x-2)= -x+2\). These rewrites can simplify sign tracking or factoring.

-Are there any special-product shortcuts here?

-A: Not really-no perfect square or sum/difference of cubes. Use distributive multiplication and combine like terms.

How do I avoid sign errors?

-A: Multiply term-by-term, write intermediate results, track each sign, and combine like terms last. Re-check by substituting one or two values.

Can synthetic division help here?

-A: Synthetic division helps test roots or divide by linear factors (e.g. test \(x=2\)), but multiplication expansion uses distributive steps instead.

How do I multiply systematically for larger polynomials?

-A: Organize in rows (like long multiplication): multiply each term of one polynomial by each term of the other, align by degree, then sum columns to combine like terms.

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