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

Answer

Multiply each term: \( (x-9)(x^2+x+2)=x^3+x^2+2x-9x^2-9x-18 = x^3-8x^2-7x-18.\)

Detailed Explanation

Problem

Expand and simplify \( (x-9)\left(x^2+x+2\right) \).

Apply the distributive property: multiply each term of the second factor by the first term, \(x\).

\(x \cdot x^2 = x^3\)

\(x \cdot x = x^2\)

\(x \cdot 2 = 2x\)
Apply the distributive property again: multiply each term of the second factor by the second term, \(-9\).

\(-9 \cdot x^2 = -9x^2\)

\(-9 \cdot x = -9x\)

\(-9 \cdot 2 = -18\)
Combine all terms obtained from the two distributions into one polynomial:

\(x^3 + x^2 + 2x – 9x^2 – 9x – 18\)
Combine like terms (group terms with the same power of \(x\)):

For the \(x^2\) terms: \(x^2 – 9x^2 = -8x^2\)

For the \(x\) terms: \(2x – 9x = -7x\)
Write the simplified polynomial in standard (descending) order:

\(x^3 – 8x^2 – 7x – 18\)

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FAQs

Q1: How do I expand \((x-9)(x^2+x+2)\)?

A1: Use distributive property: \(x(x^2+x+2)-9(x^2+x+2)=x^3+x^2+2x-9x^2-9x-18\). Combine like terms: \(\boxed{x^3-8x^2-7x-18}\).

Q2: What is the degree and leading coefficient of the expanded polynomial?

A2: The polynomial \(x^3-8x^2-7x-18\) has degree \(3\). The leading coefficient is \(1\).

Q3: What are the real roots (zeros) of \((x-9)(x^2+x+2)\)?

A3: \(x=9\) is the only real root. The quadratic \(x^2+x+2\) has discriminant \(\Delta=1-8=-7\), giving two complex roots \(\frac{-1\pm i\sqrt{7}}{2}\).

Q4: Is the polynomial fully factored over the real numbers?

A4: Yes: \((x-9)(x^2+x+2)\) is factored over \(\mathbb{R}\). Over \(\mathbb{C}\) the quadratic further factors into linear complex factors.

Q5: What is the y-intercept and x-intercepts of the graph \(y=(x-9)(x^2+x+2)\)?

A5: y-intercept: \(y(0)=(0-9)(0+0+2)=-18\). Real x-intercept: \(x=9\) only; the other two intercepts are complex (not on the real x-axis).

Q6: How do I evaluate the expression at \(x=3\)?

A6: Compute \((3-9)(3^2+3+2)=(-6)(9+3+2)=(-6)(14)=-84\).

Q7: If I divide the expanded polynomial by \((x-9)\), what is the quotient and remainder?

A7: Since the polynomial equals \((x-9)(x^2+x+2)\), dividing by \((x-9)\) yields quotient \(x^2+x+2\) and remainder \(0\).

Q8: How can I multiply a binomial by a trinomial quickly?

A8: Distribute each term of the binomial across the trinomial (or vice versa). Multiply each term and then combine like terms. This is equivalent to performing FOIL extended to three terms.

Q9: What is the sum of the coefficients of \(x^3-8x^2-7x-18\)?

A9: Sum = \(1 + (-8) + (-7) + (-18) = -32\). (-Also equals polynomial value at \(x=1\): \(-32\)).

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