Q. \( (x-4)(2x^2+5x-3) \)

Answer

Use distributive law (multiply each term of the quadratic by x and by −4):

\[
(x-4)(2x^2+5x-3)=x(2x^2+5x-3)-4(2x^2+5x-3)
\]
\[
=2x^3+5x^2-3x-8x^2-20x+12
\]
\[
=2x^3-3x^2-23x+12.
\]

Final result: \(\;2x^3-3x^2-23x+12.\)

Detailed Explanation

Problem: Multiply and simplify \( (x-4)(2x^2+5x-3) \).

Distribute \(x\) across the trinomial:
\(x\cdot(2x^2+5x-3)=2x^3+5x^2-3x\).
Distribute \(-4\) across the trinomial:
\(-4\cdot(2x^2+5x-3)=-8x^2-20x+12\).
Add the results and combine like terms:
\[
(2x^3+5x^2-3x)+(-8x^2-20x+12)
=2x^3+(5x^2-8x^2)+(-3x-20x)+12
=2x^3-3x^2-23x+12.
\]

Final answer: \(\displaystyle 2x^3-3x^2-23x+12\)

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FAQs

Q How do I expand \( (x-4)(2x^2+5x-3) \)?

A Multiply and combine like terms: \(x(2x^2+5x-3)-4(2x^2+5x-3)=2x^3-3x^2-23x+12\).

Q How do I factor \(2x^2+5x-3\) and factor the whole expression completely?

A Factor the quadratic: \(2x^2+5x-3=(2x-1)(x+3)\). So the full factorization is \((x-4)(2x-1)(x+3)\).

Q What are the zeros / x-intercepts of the polynomial?

A Set each factor to zero: \(x=4,\; x=\tfrac{1}{2},\; x=-3\).

Q What is the degree and leading coefficient of the expanded polynomial?

A Degree is 3 (cubic). Leading coefficient is 2, so the polynomial is \(2x^3-3x^2-23x+12\).

Q What is the y-intercept?

A Evaluate at \(x=0\): \(y=12\). So the y-intercept is \((0,12)\).

Q What is the end behavior of the graph?

A As \(x\to\infty\), \(f(x)\to\infty\); as \(x\to-\infty\), \(f(x)\to-\infty\). (Odd degree with positive leading coefficient.)

Q Can I use synthetic division with \(x-4\)?

A Yes. Synthetic division by 4 on \(2x^3-3x^2-23x+12\) gives quotient \(2x^2+5x-3\) and remainder 0, confirming \(x-4\) is a factor.

Q What are the multiplicities of the roots?

A Each root \(4,\tfrac{1}{2},-3\) has multiplicity 1, so the graph crosses the x-axis at each zero.

Q How many turning points can the graph have?

A A cubic can have up to 2 turning points (local max and min). Their exact locations require calculus or solving the derivative \(6x^2-6x-23\).

Q What is the domain of the polynomial function?

A The domain is all real numbers, \(\mathbb{R}\); polynomials are defined for every real \(x\).

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