Answers:
- (x-1)(x-2)(x-3)
- (x-3)²(x+3)(x+2)
- (2x-3)(x-1)(x-3)
- (x²-4)(x+3)
- (2x+3)(x+3)(x+1)(x-2)
Solutions:
1.) (x-1) ( x² - 5x + 6)
Steps:
- Find the factors of the first term.
→ The factors of the first term (x²) are x and x
- Find the factors of -12 that will give a sum which is equal to the coefficient of x (the middle term).
→ 3 + (-4) = -1
- Write the quadratic trinomial as the product of two binomials whose first terms are the factors of the first term and the second term are the pairs of factors giving the sum of the middle term.
→ (x+3) (x-4)
- The final answer is (x-1) (x+3) (x-4)
2.) (x² + x - 6 ) (x² - 6x + 9)
Steps:
- Factor (x² + x - 6 ) and (x² - 6x + 9) individually.
- Find the factors of the first term.
→ The factors of the first term (x²) are x and x
- Find the factors of -6 and 9 that will give a sum which is equal to the coefficient of x (the middle term).
→ 3 + (-2) = 1 // (-3) + (-3) = 9
- Write the quadratic trinomial as the product of two binomials whose first terms are the factors of the first term and the second term are the pairs of factors giving the sum of the middle term.
→ (x+3) (x-2) // (x-3) (x-3)
- Simplify and the final answer is (x-3)² (x+3)(x-2).
3.) (2x² - 5x + 3) (x - 3)
Steps:
- Multiply the coefficient of x²and the constant term.
→ 2(3) = 6
- Find the pairs of factors of the product.
→ 1 and 6, -1 and -6, 2 and 3, -2 and -3
- Write the coefficient of x as the sum or the difference of pairs whose sum is the coefficient of x.
→ 2x² + (-2 + -3)x + 3
- Use distributive property of multiplication over addition to rewrite the middle term.
→ 2x² -2x -3x + 3
- Use the associative property of addition to group together two terms with common factors.
→ (2x² -2x) – (3x - 3)
- Factor each group of binomials
→ 2x(x -1) -3(x-1)
- Factor the common binomial factor.
→ (2x – 3) (x-1)
- Final answer: (2x-3)(x-1)(x-3)
4.) x³ + 3x² - 4x – 12
Steps:
- Use the associative property of addition to group together two terms with common factors.
→ (x³ + 3x²) – (4x + 12)
- Factor each group of binomials
→ x² (x+3) - 4(x+3)
- Factor the common binomial factor for the final answer
→ (x²-4) (x+3)
5.) (2x+3)(x+3)(x+1)(x-2) (See attached picture for solution) .
Factoring Polynomials
Factoring a polynomial is writing it as a product of its prime factors. Finding the product of two or more numbers and expressing them as one number is multiplication. Finding the factors or numbers being multiplied and expressing one number as the product of these factors is factoring. Factoring a polynomial is similar to factoring a number. The only difference is that at least one of the factors is a polynomial.
Factoring a Quadratic Trinomial
A general quadratic trinomial in one variable is in the form “Ax2 + Bx + C” where A is not equal to 0. It is a polynomial in x whose degree is 2. The degree of a polynomial is the highest degree of its terms. To factor a general quadratic trinomial when the value of the numerical coefficient A is equal to 1, we simply find pairs of factors for C (the last term or the constant term). Then, we check which of the pair of factors will give a sum equal to B (the numerical coefficient of x or the middle term). Lastly, we write the quadratic trinomial as the product of two binomials whose first terms are the factors of the first term of the trinomial and whose second terms are the pairs of factors chosen.
Steps in factoring a general quadratic trinomial of theform Ax² + Bx + C if A=1:
- Find the factors of the constant term.
- Write the general quadratic trinomial as the product of two binomials whose first terms are x and whose second terms are the factors of the constant term whose sum is the coefficient of x or the middle term.
The following are the steps in factoring a general quadratic trinomial of the form Ax²+Bx+C if A ≠ 1:
- Multiply the coefficient of x² and the constant term.
- Find the pairs of factors of the product.
- Write the coefficient of x as the sum or the difference of the pairs of factors whose sum is the coefficient of x, the middle term.
- Use the distributive property of multiplication over addition to rewrite the middle term
- Use the associative (and commutative, if needed) property of addition to group together two terms with common factors.
- Factor each group of binomials.
- Factor the common binomial factor.
Remember:
- Remember to factor out the common monomial factor first before doing steps 1 and 2 if A = 1 in Ax² + Bx + C or steps 1-7 ifA ≠1.
- A polynomial which cannot be factored is a prime polynomial or irreducible polynomial.
To see more examples of factoring: https://brainly.ph/question/1568840
To know more about polynomials: https://brainly.ph/question/231197
To know more about quadratic equation: https://brainly.ph/question/2228648
