Determine whether (x+1) is a factor of x3+3x2+3x+1 Solve x3-6x2+11x+6=0 a Company Determine whether the…

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Determine whether (x+1) is a factor of x3+3x2+3x+1 Solve x3-6x2+11x+6=0 a Company Determine whether the profit is zero when x=4 9x+20 a Polynomial model is P(x) = x2-7x+12 determine whether x−3 is a factor

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Determine whether $(x+1)$ is a factor of $x^3 + 3x^2 + 3x + 1$ .

Let $P(x) = x^3 + 3x^2 + 3x + 1$ .
According to the Factor Theorem, $(x+1)$ is a factor if $P(-1) = 0$ .
Substitute $x=-1$ into $P(x)$ .
$P(-1) = (-1)^3 + 3(-1)^2 + 3(-1) + 1$ $P(-1) = -1 + 3(1) - 3 + 1$ $P(-1) = -1 + 3 - 3 + 1 = 0$
Since $P(-1) = 0$ , by the Factor Theorem, $(x+1)$ is a factor of $x^3 + 3x^2 + 3x + 1$ .
$\boxed{(x+1) \text{ is a factor}}$ .
1. Solve $x^3 - 6x^2 + 11x + 6 = 0$ .
Let $P(x) = x^3 - 6x^2 + 11x + 6$ .
We test for rational roots using the Rational Root Theorem. Possible rational roots are divisors of the constant term 6: $\pm 1, \pm 2, \pm 3, \pm 6$ .
Evaluate $P(x)$ for these possible roots
$P(1) = (1)^3 - 6(1)^2 + 11(1) + 6 = 1 - 6 + 11 + 6 = 12 \neq 0$
$P(-1) = (-1)^3 - 6(-1)^2 + 11(-1) + 6 = -1 - 6 - 11 + 6 = -12 \neq 0$
$P(2) = (2)^3 - 6(2)^2 + 11(2) + 6 = 8 - 24 + 22 + 6 = 12 \neq 0$
$P(-2) = (-2)^3 - 6(-2)^2 + 11(-2) + 6 = -8 - 24 - 22 + 6 = -48 \neq 0$
$P(3) = (3)^3 - 6(3)^2 + 11(3) + 6 = 27 - 54 + 33 + 6 = 12 \neq 0$
$P(-3) = (-3)^3 - 6(-3)^2 + 11(-3) + 6 = -27 - 54 - 33 + 6 = -108 \neq 0$
Since none of the possible rational roots result in $P(x)=0$ , the polynomial $x^3 - 6x^2 + 11x + 6$ has no rational roots.
Finding the exact irrational or complex roots requires methods beyond the scope of the Factor Theorem and Remainder Theorem.
$\boxed{\text{No rational roots.}}$
1. A company model the Profit by $P(x) = x^2 - 9x + 20$ . Determine whether the Profit is zero when $x=4$ .
To determine if the profit is zero when $x=4$ , substitute $x=4$ into the profit function $P(x)$ .
$P(4) = (4)^2 - 9(4) + 20$
Calculate the value of $P(4)$ .
$P(4) = 16 - 36 + 20$ $P(4) = -20 + 20$ $P(4) = 0$
Since $P(4) = 0$ , the profit is zero when $x=4$ .
$\boxed{Yes, \text{ the profit is zero when } x=4}$ .
1. A polynomial model is $P(x) = x^2 - 7x + 12$ . Determine whether $x-3$ is a factor.
According to the Factor Theorem, $(x-3)$ is a factor of $P(x)$ if $P(3) = 0$ .
Substitute $x=3$ into $P(x)$ .
$P(3) = (3)^2 - 7(3) + 12$ $P(3) = 9 - 21 + 12$ $P(3) = -12 + 12$ $P(3) = 0$
Since $P(3) = 0$ , by the Factor Theorem, $(x-3)$ is a factor of $P(x) = x^2 - 7x + 12$ .
$\boxed{Yes, (x-3) \text{ is a factor}}$ .

Point	x	y
y-intercept	0	12
x-intercept	3	0
Minimum	3.5	−0.25
x-intercept	4	0

Determine whether (x+1) is a factor of x3+3x2+3x+1 Solve x3−6x2+11x+6=0 a Company Determine whether the profit is zero when x=4 9x+20 a Polynomial model is P(x) = x2−7x+12 determine whether x-3 is a factor

Step-by-step solution

Let $P(x) = x^3 + 3x^2 + 3x + 1$ .

Substitute $x=-1$ into $P(x)$ .

Since $P(-1) = 0$ , by the Factor Theorem, $(x+1)$ is a factor of $x^3 + 3x^2 + 3x + 1$ .

Let $P(x) = x^3 - 6x^2 + 11x + 6$ .

Evaluate $P(x)$ for these possible roots

Since none of the possible rational roots result in $P(x)=0$ , the polynomial $x^3 - 6x^2 + 11x + 6$ has no rational roots.

To determine if the profit is zero when $x=4$ , substitute $x=4$ into the profit function $P(x)$ .

Calculate the value of $P(4)$ .

Since $P(4) = 0$ , the profit is zero when $x=4$ .

According to the Factor Theorem, $(x-3)$ is a factor of $P(x)$ if $P(3) = 0$ .

Substitute $x=3$ into $P(x)$ .

Since $P(3) = 0$ , by the Factor Theorem, $(x-3)$ is a factor of $P(x) = x^2 - 7x + 12$ .

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More Mathematics Questions

Determine whether (x+1) is a factor of x3+3x2+3x+1 Solve x3−6x2+11x+6=0 a Company Determine whether the profit is zero when x=4 9x+20 a Polynomial model is P(x) = x2−7x+12 determine whether x-3 is a factor

Step-by-step solution

Let P(x)=x3+3x2+3x+1P(x) = x^3 + 3x^2 + 3x + 1P(x)=x3+3x2+3x+1.

Substitute x=−1x=-1x=−1 into P(x)P(x)P(x).

Since P(−1)=0P(-1) = 0P(−1)=0, by the Factor Theorem, (x+1)(x+1)(x+1) is a factor of x3+3x2+3x+1x^3 + 3x^2 + 3x + 1x3+3x2+3x+1.

Let P(x)=x3−6x2+11x+6P(x) = x^3 - 6x^2 + 11x + 6P(x)=x3−6x2+11x+6.

Evaluate P(x)P(x)P(x) for these possible roots

Since none of the possible rational roots result in P(x)=0P(x)=0P(x)=0, the polynomial x3−6x2+11x+6x^3 - 6x^2 + 11x + 6x3−6x2+11x+6 has no rational roots.

To determine if the profit is zero when x=4x=4x=4, substitute x=4x=4x=4 into the profit function P(x)P(x)P(x).

Calculate the value of P(4)P(4)P(4).

Since P(4)=0P(4) = 0P(4)=0, the profit is zero when x=4x=4x=4.

According to the Factor Theorem, (x−3)(x-3)(x−3) is a factor of P(x)P(x)P(x) if P(3)=0P(3) = 0P(3)=0.

Substitute x=3x=3x=3 into P(x)P(x)P(x).

Since P(3)=0P(3) = 0P(3)=0, by the Factor Theorem, (x−3)(x-3)(x−3) is a factor of P(x)=x2−7x+12P(x) = x^2 - 7x + 12P(x)=x2−7x+12.