8 min readPathway sheetPart II (2 marks) by factorisation; Part III (5 marks) by long division or finding unknown coefficients

Square root of polynomials

Find a square root by factorisation (pair up equal factors) or by long division, which works like the square root of a number.

Perfect squareA polynomial equal to [p(x)]² for some polynomial p(x)

Notation√[p(x)]² = |p(x)|, the absolute value

Long division startArrange the terms in descending powers of x

Perfect square testThe long division leaves remainder zero

Which method to use

FactorisationBest when the expression is a product of factors or a single term. Example: √(49x⁶y⁴) = 7|x³y²|. Halve every power.

Long divisionBest for a long polynomial in descending powers, and for finding unknown coefficients such as a and b.

Worked example 1: by factorisation

Problem

Find the square root of (x² − 4)(x² + 5x + 6)(x² + x − 6).

Working

Factorise each part: x² − 4 = (x − 2)(x + 2).
x² + 5x + 6 = (x + 2)(x + 3).
x² + x − 6 = (x + 3)(x − 2).
Product = (x − 2)²(x + 2)²(x + 3)².
Take one factor from each pair: the square root is |(x − 2)(x + 2)(x + 3)|.

Answer

|(x − 2)(x + 2)(x + 3)|

Long division method

It works like the long division square root of a number such as 625.

1
Arrange the polynomial in descending powers of x.
2
Take the square root of the first term. Write it as the first term of the root and as the first divisor.
3
Square it, subtract, and bring down the next two terms.
4★
Double the root found so far. Divide the first term of the remainder by the first term of this double. The result is the next root term.
5
Add the new term to the doubled root to make the new divisor. Multiply the divisor by the new term and subtract.
6★
Repeat until the remainder is zero. The square root is |root|.

★ key step

Worked example 2: by long division

Problem

Find the square root of x⁴ + 4x³ + 10x² + 12x + 9.

Working

√x⁴ = x². Write x² in the root. Subtract x⁴: the remainder is 4x³ + 10x².
Double the root: 2x². Divide 4x³ by 2x²: 2x. New divisor: 2x² + 2x.
(2x² + 2x) × 2x = 4x³ + 4x². Subtract: 6x². Bring down 12x + 9 to get 6x² + 12x + 9.
Double the root x² + 2x: 2x² + 4x. Divide 6x² by 2x²: 3. New divisor: 2x² + 4x + 3.
(2x² + 4x + 3) × 3 = 6x² + 12x + 9. Subtract: the remainder is 0.
Check: (x² + 2x + 3)² = x⁴ + 4x³ + 10x² + 12x + 9.

Answer

|x² + 2x + 3|

Worked example 3: find a and b

Problem

Find a and b if 9x⁴ + 6x³ + 19x² + ax + b is a perfect square.

Working

√9x⁴ = 3x². Subtract 9x⁴: the remainder is 6x³ + 19x².
Double the root: 6x². 6x³ ÷ 6x² = x. Divisor: 6x² + x.
(6x² + x) × x = 6x³ + x². Subtract: 18x². Bring down ax + b.
Double the root 3x² + x: 6x² + 2x. 18x² ÷ 6x² = 3. Divisor: 6x² + 2x + 3.
(6x² + 2x + 3) × 3 = 18x² + 6x + 9. Remainder = (a − 6)x + (b − 9).
A perfect square leaves remainder zero, so a − 6 = 0 and b − 9 = 0.
Check: (3x² + x + 3)² = 9x⁴ + 6x³ + 19x² + 6x + 9.

Answer

a = 6 and b = 9

In real life

A square vegetable plot of area (4x² + 12x + 9) m² has side |2x + 3| m. A square root turns a known area into the length you need to fence.

Exam angle

2 marks: “Find the square root of 64x⁴ − 16x² + 1.” 64x⁴ − 16x² + 1 = (8x² − 1)², using a² − 2ab + b² = (a − b)². Square root = |8x² − 1|.

5 marks: “Find the square root of 4x⁴ − 12x³ + 13x² − 6x + 1 by long division.” First root term 2x². Double: 4x²; −12x³ ÷ 4x² = −3x. Remainder 4x² − 6x + 1; double the root: 4x² − 6x; next term 1. Remainder 0. Root = |2x² − 3x + 1|.

5 marks: “Find m and n if x⁴ − 10x³ + 27x² + mx + n is a perfect square.” Long division gives root terms x², −5x and 1, with remainder (m + 10)x + (n − 1). Set it to zero: m = −10, n = 1. Check: (x² − 5x + 1)² = x⁴ − 10x³ + 27x² − 10x + 1.

Remember it

Double, divide, add, multiply, subtract: the five moves of the long division square root, repeated until the remainder is zero.