Test: Factorising - Year 8 MCQ

To factorize the expression 3x - 9, we need to identify the greatest common factor (GCF) of the terms 3x and -9.

• The term 3x can be written as 3 * x.

• The term -9 can be written as -9 (or 3 * -3).

• The GCF of 3x and -9 is 3, as 3 is the largest number that divides both coefficients (3 and -9), and there is no x in the second term.

Factoring out 3 from 3x - 9:

• Divide 3x by 3 to get x.

• Divide -9 by 3 to get -3.

• Thus, 3x - 9 = 3(x - 3).

To verify, distribute 3 back: 3 * x = 3x and 3 * (-3) = -9, so 3(x - 3) = 3x - 9, which matches the original expression.

To find the common factor in the terms 3x and -9, we identify the factors they share:

• For 3x, the factors are 3 and x.

• For -9, the factors are -1, 3, and -3 (or we can consider the positive factor 3, as the sign is handled in factoring).

• The greatest common factor (GCF) is 3, as it is the only numerical factor common to both terms (x is not a factor of -9).

Thus, the common factor is 3.

To find the common factor in the terms x² and -5x, we identify the factors they share:

• For x², the factors are x * x.

• For -5x, the factors are -5 and x.

• The common factor is x, as it is present in both terms (the numerical coefficients 1 and -5 share no common factor other than 1).

Thus, the common factor is x.

To factorize 2x + 4, we identify the greatest common factor (GCF) of the terms 2x and 4:

• For 2x, the factors are 2 and x.

• For 4, the factors are 2 and 2 (or 4).

• The GCF is 2, as it is the largest number that divides both 2 and 4 (x is not a factor of 4).

Factoring out 2 from 2x + 4:

• Divide 2x by 2 to get x.

• Divide 4 by 2 to get 2.

• Thus, 2x + 4 = 2(x + 2).

To verify, distribute 2 back: 2 * x = 2x and 2 * 2 = 4, so 2(x + 2) = 2x + 4, which matches the original expression.

To factorize the quadratic expression x² - 9x + 8, we need to find two numbers that multiply to the constant term (8) and add to the coefficient of the x-term (-9).

• The quadratic is in the form x² + bx + c, where b = -9 and c = 8.

• We need two numbers whose product is 8 and whose sum is -9.

• Possible pairs of numbers that multiply to 8: (1, 8), (-1, -8), (2, 4), (-2, -4).

• Check their sums:

  • 1 + 8 = 9

  • -1 + (-8) = -9

  • 2 + 4 = 6

  • -2 + (-4) = -6

• The pair -1 and -8 satisfies both conditions: (-1) * (-8) = 8 and -1 + (-8) = -9.

Thus, we can write x² - 9x + 8 as (x - 1)(x - 8).

To verify, expand (x - 1)(x - 8):

• x * x = x²

• x * (-8) = -8x

• -1 * x = -x

• -1 * (-8) = 8

• Combine: x² - 8x - x + 8 = x² - 9x + 8, which matches the original expression.

The expression x² − 9 is a difference of squares, which follows the pattern a² − b² = (a − b)(a + b).

• Rewrite x² − 9 as x² − 3², where a = x and b = 3.
• Applying the difference of squares formula: x² − 9 = (x − 3)(x + 3).

To verify, expand (x − 3)(x + 3):

• x * x = x²
• x * 3 = 3x
• -3 * x = -3x
• -3 * 3 = -9
• Combine: x² + 3x − 3x − 9 = x² − 9, which matches the original expression.

To factorize 3x² + 12x, we identify the greatest common factor (GCF) of the terms 3x² and 12x:

• For 3x², the factors are 3, x, and x.

• For 12x, the factors are 12 (or 3 * 4) and x.

• The GCF is 3x, as both terms share a factor of 3 and a factor of x.

Factoring out 3x from 3x² + 12x:

• Divide 3x² by 3x to get x.

• Divide 12x by 3x to get 4.

• Thus, 3x² + 12x = 3x(x + 4).

To verify, distribute 3x back: 3x * x = 3x² and 3x * 4 = 12x, so 3x(x + 4) = 3x² + 12x, which matches the original expression.

To factorize x² - 5x, we identify the greatest common factor (GCF) of the terms x² and -5x:

• For x², the factors are x * x.

• For -5x, the factors are -5 and x.

• The GCF is x, as it is the only factor common to both terms.

Factoring out x from x² - 5x:

• Divide x² by x to get x.

• Divide -5x by x to get -5.

• Thus, x² - 5x = x(x - 5).

To verify, distribute x back: x * x = x² and x * (-5) = -5x, so x(x - 5) = x² - 5x, which matches the original expression.

The expression 4y² − 25 is a difference of squares, which follows the pattern a² − b² = (a − b)(a + b).

• Rewrite 4y² − 25 as (2y)² − 5², where a = 2y and b = 5.
• Applying the difference of squares formula: 4y² − 25 = (2y − 5)(2y + 5).

To verify, expand (2y − 5)(2y + 5):

• 2y * 2y = 4y²
• 2y * 5 = 10y
• -5 * 2y = -10y
• -5 * 5 = -25

Combine: 4y² + 10y − 10y − 25 = 4y² − 25, which matches the original expression.

To factorize 2x² − 8x, we identify the greatest common factor (GCF) of the terms 2x² and -8x:

• For 2x², the factors are 2, x, and x.

• For -8x, the factors are -8 (or 2 * -4) and x.

• The GCF is 2x, as both terms share a factor of 2 and a factor of x.

Factoring out 2x from 2x² − 8x:

• Divide 2x² by 2x to get x.

• Divide -8x by 2x to get -4.

• Thus, 2x² − 8x = 2x(x − 4).

To verify, distribute 2x back: 2x * x = 2x² and 2x * (-4) = -8x, so 2x(x − 4) = 2x² − 8x, which matches the original expression.