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AQA GCSE · Question 20 · Algebra
5x³ + ax² + bx + c ≡ kx³ + (2 - k)x² + (a² - 1)x + b/2
Work out the values of a, b and c.
5x³ + ax² + bx + c ≡ kx³ + (2 - k)x² + (a² - 1)x + b/2
Work out the values of a, b and c.
How to approach this question
The symbol '≡' means this equation is an identity, which is true for all values of x. This means the coefficient (the number in front) of each power of x must be the same on both sides.
1. Start with the highest power, x³. Set the coefficient on the left equal to the coefficient on the right to find the value of k.
2. Move to the x² term. Set the coefficients equal. Substitute the value of k you just found to solve for a.
3. Move to the x term. Set the coefficients equal. Substitute the value of a you just found to solve for b.
4. Finally, look at the constant terms (the terms without any x). Set them equal and substitute the value of b to find c.
Full Answer
This is an identity, so the coefficients of corresponding powers of x on both sides must be equal.
**Equating coefficients of x³:**
5 = k
**Equating coefficients of x²:**
a = 2 - k
Since we know k = 5:
a = 2 - 5
a = -3
**Equating coefficients of x:**
b = a² - 1
Since we know a = -3:
b = (-3)² - 1
b = 9 - 1
b = 8
**Equating constant terms:**
c = b/2
Since we know b = 8:
c = 8 / 2
c = 4
So, a = -3, b = 8, c = 4.
The statement is an identity (indicated by the ≡ symbol), which means it holds true for any value of x. For this to be possible, the coefficients of each corresponding power of x on the left-hand side (LHS) must be equal to the coefficients on the right-hand side (RHS).
Let's compare them term by term:
**For the x³ term:**
LHS coefficient: 5
RHS coefficient: k
So, **k = 5**.
**For the x² term:**
LHS coefficient: a
RHS coefficient: (2 - k)
So, a = 2 - k.
We already found that k = 5, so we can substitute this in:
a = 2 - 5
**a = -3**.
**For the x term:**
LHS coefficient: b
RHS coefficient: (a² - 1)
So, b = a² - 1.
We just found that a = -3, so we substitute this in:
b = (-3)² - 1
b = 9 - 1
**b = 8**.
**For the constant term:**
LHS coefficient: c
RHS coefficient: b/2
So, c = b/2.
We just found that b = 8, so we substitute this in:
c = 8 / 2
**c = 4**.
The values are a = -3, b = 8, and c = 4.
Common mistakes
✗ A common mistake is with negative numbers, for example calculating (-3)² as -9 instead of 9.
✗ Solving the equations in the wrong order. You must find k first, then a, then b, then c.
✗ Making a simple arithmetic error along the way.
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