To find the real zeros of the polynomial [tex]\( P(x) = x^3 - 3x^2 - 25x - 21 \)[/tex], we can use the Rational Root Theorem. This theorem states that any rational solution, or zero, of a polynomial equation with integer coefficients will be a fraction [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
In our polynomial, the constant term is [tex]\(-21\)[/tex] and the leading coefficient is [tex]\(1\)[/tex]. The factors of [tex]\(-21\)[/tex] are ±1, ±3, ±7, ±21, and the factors of [tex]\(1\)[/tex] are ±1.
Hence, the possible rational zeros of the polynomial are:
[tex]\[ \pm 1, \pm 3, \pm 7, \pm 21 \][/tex]
Let's test these values by substituting them into the polynomial [tex]\( P(x) = x^3 - 3x^2 - 25x - 21 \)[/tex].
1. Test [tex]\( x = 1 \)[/tex]:
[tex]\[
P(1) = 1^3 - 3 \cdot 1^2 - 25 \cdot 1 - 21 = 1 - 3 - 25 - 21 = -48 \quad (\text{not a zero})
\][/tex]
2. Test [tex]\( x = -1 \)[/tex]:
[tex]\[
P(-1) = (-1)^3 - 3 \cdot (-1)^2 - 25 \cdot (-1) - 21 = -1 - 3 + 25 - 21 = 0 \quad (\text{a zero})
\][/tex]
Since [tex]\( x = -1 \)[/tex] is a zero, [tex]\( (x + 1) \)[/tex] is a factor of [tex]\( P(x) \)[/tex]. Let's perform polynomial division to find the other factors.
We divide [tex]\( x^3 - 3x^2 - 25x - 21 \)[/tex] by [tex]\( x + 1 \)[/tex]:
[tex]\[ x^3 - 3x^2 - 25x - 21 \div (x + 1) \][/tex]
By performing the polynomial division (synthetic or long division), we get:
[tex]\[
(x^3 - 3x^2 - 25x - 21) ÷ (x + 1) = x^2 - 4x - 21
\][/tex]
So,
[tex]\[
P(x) = (x + 1)(x^2 - 4x - 21)
\][/tex]
Now, we need to factor the quadratic polynomial [tex]\( x^2 - 4x - 21 \)[/tex].
To factor [tex]\( x^2 - 4x - 21 \)[/tex], we look for two numbers that multiply to [tex]\(-21\)[/tex] and add to [tex]\(-4\)[/tex]. These numbers are [tex]\( -7 \)[/tex] and [tex]\( 3 \)[/tex].
Therefore, we can factor it as:
[tex]\[
x^2 - 4x - 21 = (x - 7)(x + 3)
\][/tex]
Putting it all together, the polynomial [tex]\( P(x) \)[/tex] in factored form is:
[tex]\[
P(x) = (x + 1)(x - 7)(x + 3)
\][/tex]
Thus, the real zeros are:
[tex]\[ x = -1, 7, -3 \][/tex]
To summarize:
All the real zeros of the given polynomial are:
[tex]\[ x = -1, 7, -3 \][/tex]
The polynomial in factored form is:
[tex]\[ P(x) = (x + 1)(x - 7)(x + 3) \][/tex]
