Factorize the following expressions: a) 3x − 6y + 15xy b) x3 - xy2 c) 3a2(2x-y) + 12y − 24x d) x4 - 16x2 - 36

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Here are the factorizations for each expression: a) To factorize 3x − 6y + 15xy, we find the greatest common factor (GCF) of the coefficients. Step 1: Identify the GCF of 3, 6, and 15, which is 3. 3x − 6y + 15xy = 3(x − 2y + 5xy) The factored expression is 3(x − 2y + 5xy). b) To factorize x³ − xy², we find the greatest common factor (GCF) of the terms. Step 1: Identify the GCF, which is x. x³ − xy² = x(x² − y²) Step 2: Recognize that x² − y² is a difference of squares, which factors as (x−y)(x+y). x(x² − y²) = x(x−y)(x+y) The factored expression is x(x−y)(x+y). c) To factorize 3a²(2x−y) + 12y − 24x, we first rearrange the last two terms to find a common factor. Step 1: Factor out −12 from the last two terms. 12y − 24x = −12(2x − y) Step 2: Substitute this back into the original expression. 3a²(2x−y) − 12(2x−y) Step 3: Factor out the common binomial factor (2x−y). (2x−y)(3a² − 12) Step 4: Factor out the GCF from (3a² − 12), which is 3. (2x−y)3(a² − 4) Step 5: Recognize that a² − 4 is a difference of squares, which factors as (a−2)(a+2). 3(2x−y)(a−2)(a+2) The factored expression is 3(2x−y)(a−2)(a+2). d) To factorize x⁴ − 16x² − 36, we treat it as a quadratic in terms of x². Let u = x². Step 1: Rewrite the expression in terms of u. u² − 16u − 36 Step 2: Factor the quadratic. We need two numbers that multiply to −36 and add to −16. These numbers are −18 and 2. (u − 18)(u + 2) Step 3: Substitute x² back for u. (x² − 18)(x² + 2) The factored expression is (x² − 18)(x² + 2). e) To factorize 3x² − 15x + 12, we first find the greatest common factor (GCF). Step 1: Identify the GCF of 3, 15, and 12, which is 3. 3(x² − 5x + 4) Step 2: Factor the quadratic expression x² − 5x + 4. We need two numbers that multiply to 4 and add to −5. These numbers are −1 and −4. 3(x − 1)(x − 4) The factored expression is 3(x − 1)(x − 4). f) To factorize 100x − x³, we first find the greatest common factor (GCF). Step 1: Identify the GCF, which is x. x(100 − x²) Step 2: Recognize that 100 − x² is a difference of squares, which factors as (10−x)(10+x). x(10 − x)(10 + x) The factored expression is x(10 − x)(10 + x).

Factorize the following expressions: a) 3x - 6y + 15xy b) x3 - xy2 c) 3a2(2x-y) + 12y - 24x d) x4 - 16x2 - 36

Answers

Step-by-step solution

Part a· 1 step

Identify the GCF of 3, 6, and 15, which is 3.

Part b· 2 stepsx(x-y)(x+y)

Identify the GCF, which is $x$ .

Recognize that $x^2 - y^2$ is a difference of squares, which factors as $(x-y)(x+y)$ .

Part c· 5 steps3(2x-y)(a-2)(a+2)

Factor out -12 from the last two terms.

Substitute this back into the original expression.

Factor out the common binomial factor $(2x-y)$ .

Factor out the GCF from $(3a^2 - 12)$ , which is 3.

Recognize that $a^2 - 4$ is a difference of squares, which factors as $(a-2)(a+2)$ .

Part d· 3 steps
$(x^2 - 18)(x^2 + 2)$

Rewrite the expression in terms of $u$ .

Factor the quadratic.

Substitute $x^2$ back for $u$ .

Part e· 2 steps3(x - 1)(x - 4)

Identify the GCF of 3, 15, and 12, which is 3.

Factor the quadratic expression $x^2 - 5x + 4$ .

Part f· 2 stepsx(10 - x)(10 + x)

Identify the GCF, which is $x$ .

Recognize that $100 - x^2$ is a difference of squares, which factors as $(10-x)(10+x)$ .

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More Mathematics Questions

Factorize the following expressions: a) 3x - 6y + 15xy b) x3 - xy2 c) 3a2(2x-y) + 12y - 24x d) x4 - 16x2 - 36

Answers

Step-by-step solution

Part a· 1 step

Identify the GCF of 3, 6, and 15, which is 3.

Part b· 2 stepsx(x-y)(x+y)

Identify the GCF, which is xxx.

Recognize that x2−y2x^2 - y^2x2−y2 is a difference of squares, which factors as (x−y)(x+y)(x-y)(x+y)(x−y)(x+y).

Part c· 5 steps3(2x-y)(a-2)(a+2)

Factor out -12 from the last two terms.

Substitute this back into the original expression.

Factor out the common binomial factor (2x−y)(2x-y)(2x−y).

Factor out the GCF from (3a2−12)(3a^2 - 12)(3a2−12), which is 3.

Recognize that a2−4a^2 - 4a2−4 is a difference of squares, which factors as (a−2)(a+2)(a-2)(a+2)(a−2)(a+2).

Part d· 3 steps(x2−18)(x2+2)(x^2 - 18)(x^2 + 2)(x2−18)(x2+2)

Rewrite the expression in terms of uuu.

Factor the quadratic.

Substitute x2x^2x2 back for uuu.

Part e· 2 steps3(x - 1)(x - 4)

Identify the GCF of 3, 15, and 12, which is 3.

Factor the quadratic expression x2−5x+4x^2 - 5x + 4x2−5x+4.

Part f· 2 stepsx(10 - x)(10 + x)

Identify the GCF, which is xxx.

Recognize that 100−x2100 - x^2100−x2 is a difference of squares, which factors as (10−x)(10+x)(10-x)(10+x)(10−x)(10+x).

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More Mathematics Questions

Identify the GCF, which is $x$ .

Recognize that $x^2 - y^2$ is a difference of squares, which factors as $(x-y)(x+y)$ .

Factor out the common binomial factor $(2x-y)$ .

Factor out the GCF from $(3a^2 - 12)$ , which is 3.

Recognize that $a^2 - 4$ is a difference of squares, which factors as $(a-2)(a+2)$ .

Part d· 3 steps
$(x^2 - 18)(x^2 + 2)$

Rewrite the expression in terms of $u$ .

Substitute $x^2$ back for $u$ .

Factor the quadratic expression $x^2 - 5x + 4$ .

Identify the GCF, which is $x$ .

Recognize that $100 - x^2$ is a difference of squares, which factors as $(10-x)(10+x)$ .