Q. Expand and simplify \( (2x+1)\left(-3x^{2}-x+9\right) \)

Answer

Multiply termwise:

\[
(2x+1)(-3x^2-x+9)=2x(-3x^2-x+9)+1(-3x^2-x+9).
\]

Compute each part:
\[
2x(-3x^2)=-6x^3,\quad 2x(-x)=-2x^2,\quad 2x(9)=18x,
\]
\[
1(-3x^2)=-3x^2,\quad 1(-x)=-x,\quad 1(9)=9.
\]

Combine like terms:
\[
-6x^3+(-2x^2-3x^2)+(18x-x)+9=-6x^3-5x^2+17x+9.
\]

Final result: \(\boxed{-6x^3-5x^2+17x+9}\).

Detailed Explanation

Problem

Multiply and simplify: \( (2x+1)(-3x^2 – x + 9) \)

Step-by-step solution

Distribute the binomial across each term of the trinomial
Write the product as a sum of three products:
\[
(2x+1)(-3x^2 – x + 9)
= (2x+1)(-3x^2) + (2x+1)(-x) + (2x+1)(9).
\]
Multiply each pair of factors
Compute each product separately:
\[
(2x+1)(-3x^2) = 2x\cdot(-3x^2) + 1\cdot(-3x^2) = -6x^3 – 3x^2,
\]
\[
(2x+1)(-x) = 2x\cdot(-x) + 1\cdot(-x) = -2x^2 – x,
\]
\[
(2x+1)(9) = 2x\cdot 9 + 1\cdot 9 = 18x + 9.
\]
Combine the results
Add the three expressions:
\[
-6x^3 – 3x^2 \;+\; (-2x^2 – x) \;+\; (18x + 9).
\]
Combine like terms
Group and add coefficients for each power of \(x\):
\[
-6x^3 + (-3x^2 – 2x^2) + (-x + 18x) + 9
= -6x^3 – 5x^2 + 17x + 9.
\]
Final answer
\[
(2x+1)(-3x^2 – x + 9) = -6x^3 – 5x^2 + 17x + 9.
\]

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FAQs

Q How do I expand \( (2x+1)(-3x^2-x+9) \)?

A Distribute each term: \(2x(-3x^2)=-6x^3\), \(2x(-x)=-2x^2\), \(2x(9)=18x\), \(1(-3x^2)=-3x^2\), \(1(-x)=-x\), \(1(9)=9\). Combine: \( -6x^3-5x^2+17x+9\).

Q What is the degree and leading coefficient of \( -6x^3-5x^2+17x+9 \)?

A Degree is 3 (highest power). Leading coefficient is \(-6\).

Q Can the expanded cubic \( -6x^3-5x^2+17x+9 \) be factored further?

A Possibly, but not by inspection. Use the Rational Root Theorem (±1,±3,±9,±1/2,±3/2,±9/2) and synthetic division to test roots; if none work, use numeric methods or factor over irrationals/complex numbers.

Q What common mistakes should I avoid when expanding?

A Watch signs (negative terms), multiply every term of one factor by every term of the other, and then correctly combine like terms. Don’t treat a trinomial like a binomial FOIL.

Q How can I quickly check my expanded result is correct?

A Plug a random value (e.g., \(x=1\) or \(x=2\)) into both the original product and your expanded polynomial; both should give the same number.

Q How would I solve \( (2x+1)(-3x^2-x+9)=0 \)?

A Set each factor to zero: \(2x+1=0\) gives \(x=-\tfrac{1}{2}\). For \(-3x^2-x+9=0\), use the quadratic formula \(x=\frac{-(-1)\pm\sqrt{(-1)^2-4(-3)(9)}}{2(-3)}\) to find the other roots.

Q Is FOIL applicable to \( (2x+1)(-3x^2-x+9) \)?

A Not directly: FOIL is for two binomials. Here use distributive property: multiply each term of \(2x+1\) by each term of the trinomial and sum the results.

Q What is the end-behavior of the cubic \( -6x^3-5x^2+17x+9 \)?

A Leading term \(-6x^3\) dominates: as \(x\to\infty\), \(f(x)\to -\infty\); as \(x\to -\infty\), \(f(x)\to \infty\) (odd degree, negative leading coefficient).

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