Find a polynomial with integer coefficients that satisfies the given conditions.
Q has degree 3 and zeros 
−7
 and 1 + i.

To find a polynomial \( Q(x) \) of degree 3 with integer coefficients, we need to account for the given zeros:

- \( -7 \) is one zero.
- \( 1 + i \) is another zero. 

Since the polynomial has **integer coefficients**, the complex conjugate \( 1 - i \) must also be a zero. This ensures that the coefficients remain real and integer (as non-real roots of polynomials with real coefficients occur in conjugate pairs).

### Step 1: Write the polynomial in factored form
The zeros of the polynomial are \( -7 \), \( 1 + i \), and \( 1 - i \). Therefore, the polynomial can be written as:

\[
Q(x) = (x + 7)(x - (1 + i))(x - (1 - i))
\]

### Step 2: Simplify the complex conjugate terms
First, simplify the product of \( (x - (1 + i)) \) and \( (x - (1 - i)) \). This is a difference of squares:

\[
(x - (1 + i))(x - (1 - i)) = (x - 1)^2 - (i)^2 = (x - 1)^2 + 1
\]

So, this becomes:

\[
(x - 1)^2 + 1 = x^2 - 2x + 1 + 1 = x^2 - 2x + 2
\]

### Step 3: Multiply by the remaining factor
Now, multiply this result by the remaining factor \( (x + 7) \):

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

Use distribution (expand the product):

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

\[
Q(x) = x^3 - 2x^2 + 2x + 7x^2 - 14x + 14
\]

### Step 4: Combine like terms
\[
Q(x) = x^3 + 5x^2 - 12x + 14
\]

Thus, the polynomial with integer coefficients that satisfies the given conditions is:

\[
Q(x) = x^3 + 5x^2 - 12x + 14
\]

Would you like more details on any part of the solution?

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Here are some related questions:
1. How do you find polynomials with more complicated roots?
2. What happens if a zero has multiplicity greater than 1?
3. How do you use synthetic division to verify polynomial roots?
4. How do complex conjugates affect the factorization of a polynomial?
5. How can polynomials with rational coefficients be factored?

**Tip:** Always remember that non-real roots of polynomials with real coefficients come in conjugate pairs.

Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3 and zeros −7 and 1 + i.

Learn how to find a cubic polynomial with integer coefficients given zeros −7 and 1 + i. This problem covers key algebraic concepts like complex conjugates, polynomial expansion, and the complex conjugate root theorem. Suitable for students in Grades 10-12.

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