Q. \((x-1)(x^2+3x-5)\).

Q: How do I expand (x-1)(x^2+3x-5) ?.

Distribute: x(x^2+3x-5)-1(x^2+3x-5)=x^3+3x^2-5x-x^2-3x+5. Combine like terms to get x^3+2x^2-8x+5.

Q: What is the polynomial in standard form?

Standard form orders powers descending: x^3+2x^2-8x+5.

Q: How can I check my expansion is correct?

Evaluate both expressions at a few x-values (e.g., x=0,1,2). If they match for several values, the expansion is correct. Alternatively, expand symbolically and compare.

Q: What is the degree and leading coefficient?

The degree is 3 and the leading coefficient is 1 (from x^3).

Q: What are the zeros of the polynomial?.

Solve x-1(x^2+3x-5)=0. One root is x=1; the quadratic gives x=\frac{-3\pm\sqrt{29}}{2}.

Q: What is the y-intercept of the polynomial?

Set x=0: y=5. The y-intercept is (0,5).

Answer

Problem: Expand the expression to a polynomial in standard form: left parenthesis, x, minus, 1, right parenthesis, left parenthesis, x, squared, plus, 3, x, minus, 5, right parenthesis (x−1)(x 2 +3x−5)

Multiply and combine like terms:
\[
(x-1)(x^2+3x-5)=x(x^2+3x-5)-1(x^2+3x-5)
\]
\[
= x^3+3x^2-5x -x^2-3x+5 = x^3+2x^2-8x+5.
\]

Final result: \(\;x^3+2x^2-8x+5.\)

Detailed Explanation

Problem: Expand the product \( (x-1)(x^2+3x-5) \).

Step 1 — Apply the distributive property (multiply each term of the second polynomial by each term of the first):Write the product as a sum of two products:
\( (x-1)(x^2+3x-5) = x\,(x^2+3x-5) – 1\,(x^2+3x-5) \).
Step 2 — Multiply the first term \(x\) by each term of the second polynomial:Compute:
\( x\cdot x^2 = x^3 \),
\( x\cdot 3x = 3x^2 \),
\( x\cdot (-5) = -5x \).

So \( x\,(x^2+3x-5) = x^3 + 3x^2 – 5x \).
Step 3 — Multiply the second term \(-1\) by each term of the second polynomial:Compute:
\( -1\cdot x^2 = -x^2 \),
\( -1\cdot 3x = -3x \),
\( -1\cdot (-5) = +5 \).

So \( -1\,(x^2+3x-5) = -x^2 – 3x + 5 \).
Step 4 — Add the two results and combine like terms:Sum:
\( (x^3 + 3x^2 – 5x) + (-x^2 – 3x + 5) \).

Group like terms:
Cubic term: \( x^3 \).
Quadratic terms: \( 3x^2 + (-x^2) = 2x^2 \).
Linear terms: \( -5x + (-3x) = -8x \).
Constant term: \( 5 \).

Thus the combined result is
\( x^3 + 2x^2 – 8x + 5 \).
Final Answer:\( x^3 + 2x^2 – 8x + 5 \)

See full solution

Solve homework faster - try our AI homework help tools now!

Explain

FAQs

How do I expand \( (x-1)(x^2+3x-5) \)?.

Distribute: \(x(x^2+3x-5)-1(x^2+3x-5)=x^3+3x^2-5x-x^2-3x+5\). Combine like terms to get \(x^3+2x^2-8x+5\).

What is the polynomial in standard form?

Standard form orders powers descending: \(x^3+2x^2-8x+5\).

How can I check my expansion is correct?

Evaluate both expressions at a few \(x\)-values (e.g., \(x=0,1,2\)). If they match for several values, the expansion is correct. Alternatively, expand symbolically and compare.

What is the degree and leading coefficient?

The degree is 3 and the leading coefficient is 1 (from \(x^3\)).

What are the zeros of the polynomial?.

Solve \(x-1\)(\(x^2+3x-5\))\(=0\). One root is \(x=1\); the quadratic gives \(x=\frac{-3\pm\sqrt{29}}{2}\).

Can the polynomial be factored further over the rationals?

What is a quick method to multiply expressions like this?

Use the distributive property: multiply each term of the first factor by each term of the second, then combine like terms. For binomial×trinomial do three multiplications and sum.

What is the \(y\)-intercept of the polynomial?

Set \(x=0\): \(y=5\). The y-intercept is \( (0,5)\).

Math AI tools solve different problems.
Find your favorite today!

173,935+ happy customers
Math, Calculus, Geometry, etc.

✅ AI math helper 👤 AI geometry solver ✏️ AI calculus help