Q. \( (4x+5)\bigl(x^{2}-2x+5\bigr) \)
Answer
We interpret the expression as \((4x+5)(x^2-2x+5)\). Distribute:
\[
(4x+5)(x^2-2x+5)=4x(x^2-2x+5)+5(x^2-2x+5)
\]
\[
=4x^3-8x^2+20x+5x^2-10x+25
\]
\[
=4x^3-3x^2+10x+25.
\]
Final result: \(\boxed{4x^3-3x^2+10x+25}\).
Detailed Explanation
ProblemMultiply: \( (4x+5)\bigl(x^{2}-2x+5\bigr) \)
Step 1 – Distribute each term
Multiply \(4x\) by the trinomial and \(5\) by the trinomial separately:
\[
4x\cdot\bigl(x^{2}-2x+5\bigr)
=4x\cdot x^{2}+4x\cdot(-2x)+4x\cdot5
=4x^{3}-8x^{2}+20x
\]
\[
5\cdot\bigl(x^{2}-2x+5\bigr)
=5x^{2}-10x+25
\]
Step 2 – Add the results and combine like terms
Now add the two polynomials term by term:
\[
(4x^{3}-8x^{2}+20x)+(5x^{2}-10x+25)
=4x^{3}+(-8x^{2}+5x^{2})+(20x-10x)+25
=4x^{3}-3x^{2}+10x+25
\]
Answer
\[
(4x+5)(x^{2}-2x+5)=4x^{3}-3x^{2}+10x+25
\]
See full solution
FAQs
How do I expand \( (4x+5)(x^2-2x+5) \)?
A: Distribute each term: \(4x(x^2-2x+5)+5(x^2-2x+5)=4x^3-8x^2+20x+5x^2-10x+25\), so the expanded form is \(4x^3-3x^2+10x+25\).
What is the degree and leading coefficient of the product?
A: The polynomial \(4x^3-3x^2+10x+25\) has degree 3 and leading coefficient 4.
How can I factor the cubic back into its given factors?
A: It’s already factored as \( (4x+5)(x^2-2x+5) \). The quadratic has no real linear factors, so this is the factorization over the rationals.
What are the real and complex roots?
A: The linear factor gives real root \(x=-\tfrac{5}{4}\). For \(x^2-2x+5\), use quadratic formula: roots are \(x=1\pm 2i\) (complex conjugates).
How can I check a root like \(x=-\tfrac{5}{4}\) quickly?
A: Substitute into a factor: \(4(-\tfrac{5}{4})+5= -5+5=0\). If a factor equals zero, the whole product is zero, so \(x=-\tfrac{5}{4}\) is a root.
How do I differentiate \( (4x+5)(x^2-2x+5) \)?
A: Use product rule or differentiate the expanded form. Expanded derivative: \(\frac{d}{dx}(4x^3-3x^2+10x+25)=12x^2-6x+10\).
How does the graph behave as \(x\to\pm\infty\)?
A: Leading term \(4x^3\) dominates, so as \(x\to\infty\), \(f(x)\to\infty\); as \(x\to-\infty\), \(f(x)\to-\infty\). With one real root, the curve crosses the x-axis once.
What distribution/FOIL steps are used to avoid mistakes?
A: Multiply each term of one factor by each of the other: \(4x\cdot x^2,\,4x\cdot(-2x),\,4x\cdot5,\,5\cdot x^2,\,5\cdot(-2x),\,5\cdot5\). Then combine like terms carefully to prevent sign or coefficient errors.
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