Answer :
To factor the polynomial [tex]\( f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 \)[/tex] completely, we can break it down into simpler expressions. Here is a step-by-step guide on how to achieve that:
1. Look for Rational Roots:
We start by applying the Rational Root Theorem to look for possible rational roots by testing factors of the constant term (70) with the leading coefficient (1).
2. Test Potential Roots:
Check possible roots: [tex]\( \pm 1, \pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70 \)[/tex]. Test them in the polynomial by direct substitution or synthetic division to see if they yield zero.
3. Find Actual Roots:
After testing, you will find that [tex]\( x = 5 \)[/tex] and [tex]\( x = -2 \)[/tex] are roots. Thus, [tex]\( (x - 5) \)[/tex] and [tex]\( (x + 2) \)[/tex] are factors of the polynomial.
4. Divide Polynomial:
Perform polynomial division, dividing the original polynomial by [tex]\( (x - 5)(x + 2) \)[/tex] step-by-step, to break it down further. This will give you a quadratic polynomial as the quotient, [tex]\( x^2 - 7 \)[/tex].
5. Check Factorization:
Now the polynomial factors completely into:
[tex]\[
f(x) = (x - 5)(x + 2)(x^2 - 7)
\][/tex]
6. Confirm Your Factors:
Ensure that when you multiply these back together, they expand to the original polynomial.
Therefore, the complete factorization of the polynomial [tex]\( f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 \)[/tex] is [tex]\( (x - 5)(x + 2)(x^2 - 7) \)[/tex].
1. Look for Rational Roots:
We start by applying the Rational Root Theorem to look for possible rational roots by testing factors of the constant term (70) with the leading coefficient (1).
2. Test Potential Roots:
Check possible roots: [tex]\( \pm 1, \pm 2, \pm 5, \pm 7, \pm 10, \pm 14, \pm 35, \pm 70 \)[/tex]. Test them in the polynomial by direct substitution or synthetic division to see if they yield zero.
3. Find Actual Roots:
After testing, you will find that [tex]\( x = 5 \)[/tex] and [tex]\( x = -2 \)[/tex] are roots. Thus, [tex]\( (x - 5) \)[/tex] and [tex]\( (x + 2) \)[/tex] are factors of the polynomial.
4. Divide Polynomial:
Perform polynomial division, dividing the original polynomial by [tex]\( (x - 5)(x + 2) \)[/tex] step-by-step, to break it down further. This will give you a quadratic polynomial as the quotient, [tex]\( x^2 - 7 \)[/tex].
5. Check Factorization:
Now the polynomial factors completely into:
[tex]\[
f(x) = (x - 5)(x + 2)(x^2 - 7)
\][/tex]
6. Confirm Your Factors:
Ensure that when you multiply these back together, they expand to the original polynomial.
Therefore, the complete factorization of the polynomial [tex]\( f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 \)[/tex] is [tex]\( (x - 5)(x + 2)(x^2 - 7) \)[/tex].
