Gr12 Mathematics: Ch 5.10 Factor Theorem

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What is the condition for cx - d to be a factor of p(x) according to the Factor Theorem?

If p(d/c) is a negative number

If p(d/c) is a positive number

If p(d/c) is a non-zero constant

If p(d/c) is zero (correct)

What can be concluded if p(d/c) = 0?

cx - d is a factor of p(x) (correct)

p(x) is a quadratic polynomial

p(x) is a linear polynomial

cx - d is not a factor of p(x)

What is the general form of a polynomial p(x) that has a factor cx - d?

p(x) = (cx - d) \* Q(x) (correct)

p(x) = Q(x) / (cx - d)

p(x) = cx - d + Q(x)

p(x) = cx - d / Q(x)

What is the relationship between the remainder and the Factor Theorem?

The remainder is zero if cx - d is a factor of p(x) Signup and view all the answers

What is a useful application of the Factor Theorem?

Factorizing cubic polynomials by finding one factor and then using polynomial division to find the remaining factors Signup and view all the answers

What can be concluded if p(x) has a root d/c?

cx - d is a factor of p(x) Signup and view all the answers

If p(x) is a polynomial and cx - d is a factor of p(x), what can be concluded about p(d/c)?

p(d/c) is equal to zero Signup and view all the answers

If p(x) is a polynomial and p(d/c) = 0, what can be concluded about cx - d?

cx - d is a factor of p(x) Signup and view all the answers

What is the polynomial Q(x) in the expression p(x) = (cx - d)Q(x)?

the quotient polynomial Signup and view all the answers

Why is the Factor Theorem particularly useful for factorizing cubic polynomials?

because it provides a method to find one factor and then use polynomial division to find the remaining factors Signup and view all the answers

What is the relationship between the roots of a polynomial p(x) and its factors?

if x = d/c is a root of p(x), then cx - d is a factor of p(x) Signup and view all the answers

What is a special case of the Remainder Theorem?

the Factor Theorem Signup and view all the answers

If $p(x) = x^3 + 2x^2 - 7x - 12$ and $p(-2) = 0$, what can be concluded about the factorization of $p(x)$?

$x - 2$ is a factor of $p(x)$ Signup and view all the answers

If $p(x) = x^2 - 4x - 3$ and $cx - d$ is a factor of $p(x)$, what can be concluded about the value of $c$ and $d$?

$c = 1$ and $d = 3$ Signup and view all the answers

If $p(x) = x^4 - 2x^3 - 3x^2 + 6x + 1$ and $p(1) = 0$, what can be concluded about the factorization of $p(x)$?

$x - 1$ is a factor of $p(x)$ Signup and view all the answers

If $p(x) = 2x^3 + 5x^2 - 3x - 1$ and $cx - d$ is a factor of $p(x)$, what is the degree of the quotient polynomial $Q(x)$?

$1$ Signup and view all the answers

If $p(x) = x^3 - 2x^2 - 5x + 6$ and $p(-3) = 0$, what can be concluded about the value of $p(-3/2)$?

$0$ Signup and view all the answers

If $p(x) = x^4 - 4x^3 + 7x^2 - 12x + 9$ and $cx - d$ is a factor of $p(x)$, what can be concluded about the value of $c$ and $d$?

$c = 1$ and $d = -1$ Signup and view all the answers