Q. Multiply and simplify: \( (2x – 3)(3x^2 + x – 4) \)

Answer

Multiply term-by-term:

\[
(2x-3)(3x^2+x-4)=2x(3x^2+x-4)-3(3x^2+x-4)
\]
\[
=6x^3+2x^2-8x-9x^2-3x+12
\]
Combine like terms:
\[
=6x^3-7x^2-11x+12.
\]

Final result: \(6x^3-7x^2-11x+12\).

Detailed Explanation

Expand and simplify the product \( (2x-3)(3x^2+x-4) \).

Use the distributive property (multiply each term of the first factor by the entire second factor):

First multiply \(2x\) by each term of \(3x^2+x-4\):

\[
2x\cdot(3x^2+x-4)=2x\cdot3x^2+2x\cdot x+2x\cdot(-4)
=6x^3+2x^2-8x
\]
Now multiply \(-3\) by each term of \(3x^2+x-4\):

\[
-3\cdot(3x^2+x-4)=-3\cdot3x^2-3\cdot x-3\cdot(-4)
=-9x^2-3x+12
\]
Add the two results and combine like terms:

\[
(6x^3+2x^2-8x)+(-9x^2-3x+12)
=6x^3+(2x^2-9x^2)+(-8x-3x)+12
=6x^3-7x^2-11x+12
\]

Final answer: \(\;6x^3-7x^2-11x+12\)

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FAQs

Q What is the expanded form of \( (2x-3)(3x^2+x-4) \)?

A Multiply each term: \(2x\cdot3x^2=6x^3\), \(2x\cdot x=2x^2\), \(2x\cdot(-4)=-8x\), \(-3\cdot3x^2=-9x^2\), \(-3\cdot x=-3x\), \(-3\cdot(-4)=12\). Combine: \(6x^3-7x^2-11x+12\).

Q Which method is best: FOIL or distributive?

A Use the distributive property (multiply each term of the binomial by each term of the trinomial). FOIL is for binomial×binomial; distributive works universally and avoids missed terms.

Q Can the cubic be factored further?

A Yes. \(3x^2+x-4=(3x+4)(x-1)\). So the full factorization is \((2x-3)(3x^2+x-4)=(2x-3)(x-1)(3x+4)\).

Q What are the roots/zeros of the polynomial?

A Solve \((2x-3)(x-1)(3x+4)=0\). Roots: \(x=\tfrac{3}{2},\; x=1,\; x=-\tfrac{4}{3}\).

Q What is the degree and leading coefficient?

A The polynomial is degree 3 (cubic). The leading coefficient is 6 (from \(6x^3\)).

Q What is the end behavior of the function \(f(x)=6x^3-7x^2-11x+12\)?

A For odd degree with positive leading coefficient: as \(x\to\infty\), \(f(x)\to\infty\); as \(x\to-\infty\), \(f(x)\to-\infty\).

Q Common mistakes to avoid when expanding?

A Forgetting to multiply every term, sign errors (especially with negatives), and failing to combine like terms correctly. Check by re-expanding or factoring the expanded result to confirm.

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