Answer :
Certainly! Let's factor the polynomial [tex]\(x^4 + 17x^2 + 70\)[/tex] completely.
### Step-by-Step Solution
1. Understanding the Polynomial:
- The given polynomial is [tex]\(x^4 + 17x^2 + 70\)[/tex].
- Notice that this can be expressed as a quadratic in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex]. Substituting for [tex]\(y\)[/tex], we have:
[tex]\[
y^2 + 17y + 70
\][/tex]
- Now, the polynomial looks like a standard quadratic equation in [tex]\(y\)[/tex].
2. Factoring the Quadratic:
- We need two numbers that multiply to 70 and add up to 17.
- The numbers 7 and 10 satisfy [tex]\( (7 \times 10 = 70) \)[/tex] and [tex]\( (7 + 10 = 17) \)[/tex].
3. Writing the Factors:
- Using the numbers found above, we can write the quadratic in its factored form as:
[tex]\[
(y + 7)(y + 10)
\][/tex]
- Replace [tex]\(y\)[/tex] with [tex]\(x^2\)[/tex] to return to the original variable:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]
4. Conclusion:
- The polynomial [tex]\(x^4 + 17x^2 + 70\)[/tex] factors completely to:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]
Therefore, the factors of the polynomial with integer coefficients are [tex]\((x^2 + 7)\)[/tex] and [tex]\((x^2 + 10)\)[/tex].
### Step-by-Step Solution
1. Understanding the Polynomial:
- The given polynomial is [tex]\(x^4 + 17x^2 + 70\)[/tex].
- Notice that this can be expressed as a quadratic in terms of [tex]\(x^2\)[/tex]. Let [tex]\(y = x^2\)[/tex]. Substituting for [tex]\(y\)[/tex], we have:
[tex]\[
y^2 + 17y + 70
\][/tex]
- Now, the polynomial looks like a standard quadratic equation in [tex]\(y\)[/tex].
2. Factoring the Quadratic:
- We need two numbers that multiply to 70 and add up to 17.
- The numbers 7 and 10 satisfy [tex]\( (7 \times 10 = 70) \)[/tex] and [tex]\( (7 + 10 = 17) \)[/tex].
3. Writing the Factors:
- Using the numbers found above, we can write the quadratic in its factored form as:
[tex]\[
(y + 7)(y + 10)
\][/tex]
- Replace [tex]\(y\)[/tex] with [tex]\(x^2\)[/tex] to return to the original variable:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]
4. Conclusion:
- The polynomial [tex]\(x^4 + 17x^2 + 70\)[/tex] factors completely to:
[tex]\[
(x^2 + 7)(x^2 + 10)
\][/tex]
Therefore, the factors of the polynomial with integer coefficients are [tex]\((x^2 + 7)\)[/tex] and [tex]\((x^2 + 10)\)[/tex].
