Q. Expand \( (-x+4)(3x^2-2x-7) \)

Answer

Compute by distributing:
\[
(-x+4)(3x^2-2x-7)=-x(3x^2-2x-7)+4(3x^2-2x-7)
\]
\[
=-3x^3+2x^2+7x+12x^2-8x-28=-3x^3+14x^2-x-28.
\]

Final answer: \(-3x^3+14x^2-x-28\).

Detailed Explanation

Problem: \( (-x+4)(3x^2-2x-7) \)

  1. Distribute the first term \(-x\):

    Compute \(-x\) times each term of \(3x^2-2x-7\):

    \[
    -x(3x^2-2x-7) = -3x^3 + 2x^2 + 7x
    \]

  2. Distribute the second term \(4\):

    Compute \(4\) times each term of \(3x^2-2x-7\):

    \[
    4(3x^2-2x-7) = 12x^2 – 8x – 28
    \]

  3. Add the results and combine like terms:

    \[
    (-3x^3 + 2x^2 + 7x) + (12x^2 – 8x – 28)
    = -3x^3 + (2x^2+12x^2) + (7x-8x) – 28
    \]

    \[
    = -3x^3 + 14x^2 – x – 28
    \]

Final answer: \(\displaystyle -3x^3 + 14x^2 – x – 28\)

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Explain It

FAQs

How do I expand \(({-}x+4)(3x^{2}-2x-7)\)?

-A: Use distributive property: multiply each term in the binomial by each term in the trinomial. Result: \(-3x^{3}+14x^{2}-x-28\).

Show the step-by-step multiplication.

-A: \((-x)(3x^{2}-2x-7)=-3x^{3}+2x^{2}+7x\). \(4(3x^{2}-2x-7)=12x^{2}-8x-28\). -Add: \(-3x^{3}+(2+12)x^{2}+(7-8)x-28=-3x^{3}+14x^{2}-x-28\).

Can I use FOIL here?

-A: FOIL applies only to two binomials. For a binomial times a trinomial use distributive property (multiply each binomial term by all three trinomial terms), which is the same idea extended.

What mistakes should I watch for?

-A: Watch signs when multiplying negatives, combine like terms correctly, and ensure each term from one factor multiplies every term of the other. Recheck by substituting a value for \(x\).

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

-A: Degree is 3 (highest power \(x^{3}\)). Leading coefficient is \(-3\) (coefficient of \(x^{3}\)).

How many terms does the expanded polynomial have?

-A: Four terms after combining like terms: \(-3x^{3}+14x^{2}-x-28\).

Can this polynomial be factored further over the integers?

-A: No. The factor \(3x^{2}-2x-7\) has discriminant \(88\), not a perfect square, so it has no rational-factorization; the original product already shows the integer factorization up to a sign.

Is there a simpler form using a sign change?

-A: Yes: \(-x+4=-(x-4)\). So the product equals \(-(x-4)(3x^{2}-2x-7)\), which can be useful for sign checks.

How can I check my expansion quickly?

-A: Substitute a random value, e.g. \(x=1\): original gives \((-1+4)(3-2-7)=3(-6)=-18\). Expanded gives \(-3+14-1-28=-18\). Matching values confirm correctness.

What are the roots of the polynomial \(-3x^{3}+14x^{2}-x-28\)?

-A: Roots include the roots of \(3x^{2}-2x-7=0\): \(x=\dfrac{1\pm\sqrt{22}}{3}\). The third root comes from \(-x+4=0\), i.e. \(x=4\).

How to multiply efficiently without sign errors?

-A: Rewrite \(-x+4\) as \(4-x\) or \(-(x-4)\) if helpful. Multiply term-by-term, write intermediate sums, and combine like terms. Keep track of signs next to each term instead of mentally flipping them.

Can I use polynomial long multiplication or a table?

-A: Yes. -A multiplication table (grid) or column method helps organize products and reduces sign/term-miss errors. Both are equivalent to distributive multiplication and useful for longer polynomials.
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