Factorising An Algebraic Expression - National 5 Maths Revision - BBC

In this guide

1. Revise
2. Test

Pages

1. Factorising by finding a common factor
2. Difference of two squares
3. Factorising trinomials
4. Factorising trinomials: extension

Factorising trinomials: extension

Coefficient for x2 greater than 1

When the coefficient for \({x^2}\) is greater than 1, there is a different method to follow.

Here is one method.

Remember \(a{x^2} + bx + c\)

Step 1:\(a \times c\) gives the number needed to find factors that will also add to give b.

Step 2: Once we have the correct factors, replace \(bx\) with these factors.

Step 3: Take a common factor of the first two terms then do the same for the last two terms.

Step 4: You should notice that you have two brackets the same. One bracket is for your answer and the other bracket contains the common factors in front of the brackets.

Example

Factorise \(3{x^2} - 7x - 6\)

Answer

\(a = 3,b = - 7,c = - 6\)

Step 1

\(a \times c = 3 \times - 6 = - 18\)

We need factors of -18 which add to give -7.

Factors of -18	Adding the factors
\(1x - 18\)	\(- 17\)
\(-1 x 18\)	\(17\)
\(2x - 9\)	\(- 7\)
\(-2 x 9\)	\(7\)
\(3x - 6\)	\(- 3\)
\(-3 x 6\)	\(3\)

Factors of -18	\(1x - 18\)
Adding the factors	\(- 17\)

Factors of -18	\(-1 x 18\)
Adding the factors	\(17\)

Factors of -18	\(2x - 9\)
Adding the factors	\(- 7\)

Factors of -18	\(-2 x 9\)
Adding the factors	\(7\)

Factors of -18	\(3x - 6\)
Adding the factors	\(- 3\)

Factors of -18	\(-3 x 6\)
Adding the factors	\(3\)

Required factors are 2 and -9.

Step 2

Replace \(7x\) with \(+2x\) and \(-9x\) in the order which helps to get common factors.

\(3x^{2}-9x+2x-6\)

Step 3

\(3x(x-3) +2(x-3)\)

(Take a common factor of the first two terms, then the last two terms)

Step 4

\((x - 3)(3x + 2)\)

Question

Factorise the following:

\(4{x^2} + 8x - 5\)

Show answer

\(a = 4,b = 8,c = - 5\)

Step 1

\(a \times c = 4 \times - 5 = - 20\)

Factors of -20	Adding the factors
\(1x - 20\)	\(- 19\)
\(-1 x 20\)	\( 19\)
\(2x-10\)	\(-8\)
\(-2 x 10\)	\(8\)
\(4x -5\)	\(-1\)
\(-4x5\)	\(1\)

Factors of -20	\(1x - 20\)
Adding the factors	\(- 19\)

Factors of -20	\(-1 x 20\)
Adding the factors	\( 19\)

Factors of -20	\(2x-10\)
Adding the factors	\(-8\)

Factors of -20	\(-2 x 10\)
Adding the factors	\(8\)

Factors of -20	\(4x -5\)
Adding the factors	\(-1\)

Factors of -20	\(-4x5\)
Adding the factors	\(1\)

Required factors are -2 and 10.

Step 2

Replace +8x with -2x and +10x in the order which helps to get common factors.

\(4{x^2} - 2x + 10x - 5\)

(Replace)

Step 3

\(2x(2x - 1) + 5(2x - 1)\)

(Common factors)

Step 4

\((2x - 1)(2x + 5)\)

Question

Factorise \(10{x^2} + 38x - 8\)

Show answer

Here, notice that this expresion can be factorised by first taking out a common factor of 2.

\(= 2(5{x^2} + 19x - 4)\)

Step 1

\(a = 5,b = 19,c = - 4\)

\(a \times c = 5 \times - 4 = - 20\)

Using the factors from the previous question which have to add to 19 here, we choose -1 and 20.

Step 2

Replace +19x with -x and +20x in the order which helps to get common factors.

\(2(5{x^2} - x + 20x - 4)\)

(Replace)

Step 3

\(2[x(5x - 1) + 4(5x - 1)]\)

(Common factors)

Step 4

\(2(5x - 1)(x + 4)\)

Next upTest your understandingPrevious pageFactorising trinomials

More guides on this topic

• Expansion of brackets
• Completing the square in a quadratic expression
• Simplifying algebraic fractions
• Applying the four operations to algebraic fractions
• Determining the equation of a straight line
• Working with linear equations and inequations
• Working with simultaneous equations
• Changing the subject of a formula
• Determine the equation of a quadratic function from its graph
• Sketching a quadratic function
• Identifying features of a quadratic function
• Solving a quadratic equation
• Solving a quadratic equation using the quadratic formula
• Using the discriminant to determine the number of roots
• Video playlist

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