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Math Algebra Polynomial Factor Theorem Factor Theorem Factor Theorem Definition
Factor Theorem
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Zero Factor Theorem
Polynomial is a very important topic in mathematics. Polynomials are used everywhere in maths. Polynomials are an expression of
Factor Theorem Proof
the form -
Factor Theorem Examples
Where, n is a positive integer and
.
There are two important theorems that play a vital role while dealing with polynomials. One is remainder theorem and other is
Related Concepts
factor theorem. Factor theorem is derived from the remainder theorem. Let us first learn about the remainder theorem, according
Polynomial Factor Theorem
If P(x) be a polynomial and it is divided by another polynomial (x - a), then the quotient is represented by Q(x) and the remainder
Prime Factorization Theorem
to which is given by R(x). Then remainder theorem says that P(x) = (x - a) Q(x) + R(x)
Factor Apothem Theorem
Factor theorem is a special case of remainder theorem. Factor theorem states that if a polynomial P(x) is evenly divided by another
Baye's Theorem
polynomial (x - a), then it leaves no remainder; i.e. R(x) = 0 and hence (x - a) is said to be the factor of P(x). According to factor Binomial Theorems
theorem -
Bisector Theorem
P(x) = (x - a) Q(x)
Calculus Theorems
In other words, (x - a) is said to be the factor of polynomial P(x); if P(a) = 0.
Related Formulas
This theorem tells us that there is a relation between factors of the polynomial and zeros of the polynomial. This theorem makes us find the roots of the polynomials and also helps us to solve the polynomials of higher degree.
Equation by Factoring Examples of Pythagorean Theorem Problems
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Pythagorean Theorem Triples Formula The Central Limit Theorem States
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Factor Theorem Definition
Factor by Grouping Factor Tree
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The factor theorem is a theorem commonly applied to factorizing and finding the roots of polynomial equations. If P(x), a polynomial in x is divided by x - a and the remainder is zero, then (x - a) is a factor of P(x).
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P(x) = (x - a) Q(x)
Factoring Worksheets
Zero Factor Theorem
Factors Worksheet
Zero factor theorem is the inverse case of the factor theorem. It is used to find the factors of the polynomial equations. This
Central Limit Theorem Worksheet Converse of the Pythagorean Theorem Worksheet
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theorem states that, if x - r is the factor of P(x), then r is a zero of the polynomial P(x).
Solved Example Question: Find the roots of the x 2 + x - 6 = 0 Solution: Given polynomial equation is, x 2 + x - 6 = 0 x 2 + x - 6 = 0 => x 2 + 3x - 2x - 6 = 0 => x(x + 3) - 2(x - 3) = 0 => (x - 2)(x + 3) = 0 Product of two binomials is zero. So, either x - 2 = 0 or x + 3 = 0 => x = 2 or x = -3 Therefore, roots of the given polynomial equation are 2, -3.
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Factor Theorem Proof
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The polynomial P(x) is exactly divisible by the x - a iff the value of the polynomial is zero for x = a. Proof: Since P(x) is divided by x - a P(x) = (x - a) . Q(x) + P(a) ( by remainder theorem ) (Dividend = Divisor x quotient + Remainder Division Algorithm) But, P(a) = 0 is given. => P(x) = (x - a) . Q(x) => (x - a) is a factor of P(x). Converse: If x - a is a factor of P(x), then P(a) = 0. P(x) = (x - a) . Q(x) + P(a) If (x - a) is a factor, then the remainder should be zero (x - a divides P(x) exactly) P(a) = 0
Factor Theorem Examples
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Given below are some of the examples on factor theorem.
Solved Examples Question 1: Determine whether x - 2 is a factor of x 2 - 7x + 10. Solution: Let P(x) = x 2 - 7x + 10 By the factor theorem, x - 2 is the factor of P(x) if P(2) = 0. Find the value of P(2). P(x) = x 2 - 7x + 10 P(2) = 2 2 - 7 * 2 + 10 (Substitute x = 2) = 4 - 14 + 10 = 0 Since P(2) = 0 => x - 2 is a factor of P(x).
Question 2: Determine whether x - 3 is a factor of x 3 - 3x 2 + 4x - 12. Solution: Let P(x) = x 3 - 3x 2 + 4x - 12 By the factor theorem, x - 3 is the factor of P(x) if P(3) = 0. Find the value of P(3). P(x) = x 3 - 3x 2 + 4x - 12 P(3) = 3 3 - 3 * 3 2 + 4 * 3 - 12 (Substitute x = 3) = 27 - 27 + 12 - 12 = 0 => x - 3 is a factor of P(x).
Question 3: Show that x + 1 is a factor of 2x 3 + 5x 2 - 9x - 12. Solution: Let P(x) = 2x 3 + 5x 2 - 9x - 12 By the factor theorem, x - (-1) is the factor of P(x) if P(-1) = 0. Find the value of P(-1). P(x) = 2x 3 + 5x 2 - 9x - 12 P(x) = 2(-1)3 + 5(-1)2 - 9(-1) - 12 = -2 + 5 + 9 - 12 = 0 => x + 1 is a factor of P(x).
Question 4: Find the value of a so that x 4 + 2x 3 - ax 2 + x - 2 has (x + 2) as its factor. Solution: Let P(x) = x 4 + 2x 3 - ax 2 + x - 2 Since x + 2 is a factor of P(x), so P(-2) = 0 => P(-2) = (-2)4 + 2(-2)3 - a(-2)2 + (-2) - 2 = 0 (substitute x = -2) => 16 - 16 - 4a - 2 - 2 = 0 => -4a - 4 = 0 => -4a = 4 => a = -1 So, the value of a is -1.
Question 5: Factorize 2x 2 + x - 3 using factor theorem. Solution: Let P(x) = 2x 2 + x - 3 The factors of the constant 3 are +1, -1, +3, -3. Put x = 1 in P(x) => P(1) = 2 + 1 - 3 = 0 => x - 1 is one of a factor of P(x). Now, divide 2x 2 + x - 3 by x - 1 to get the other factor. 2x 2 + x - 3 = (2x + 3)(x - 1)
Question 6: If (x - 2) and (x - 3) are factors of x 3 + ax 2 + bx + 12, find a and b. Solution: Let P(x) = x 3 + ax 2 + bx + 12 Step 1: Case 1: Since x - 2 is a factor of P(x), so P(2) = 0 => 2 3 + a * 2 2 + b * 2 + 12 = 0 => 8 + 4a + 2b + 12 = 0 => 4a + 2b + 20 = 0 2a + b + 10 = 0 .............................(i) Case 2: Since x - 3 is a factor of P(x), so P(3) = 0 => 3 3 + a * 3 2 + b * 3 + 12 = 0 => 27 + 9a + 3b + 12 = 0 => 9a + 3b + 39 = 0 3a + b + 13 = 0 ................................(ii) Step 2: Subtracting equation (i) from (ii), we get, => 3a + b + 13 - (2a + b + 10) = 0 => 3a - 2a + b - b + 13 - 10 = 0 => a = -3 Substitute a = -3 in equation (1), 2a + b + 10 = 0 -6 + b = -10 b = -4
Zeros of Polynomials
Remainder Theorem
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