Expanding a quartic expression after substitution

Key takeaway: This problem checks whether you can substitute a value into a polynomial expression and apply exponent rules before simplifying. The key is to evaluate the power first, then handle subtraction and division in the correct order.

Key idea

When you are asked to evaluate an expression at a given value, replace the variable first and then simplify using order of operations.

For this expression:

$\frac{1-x^4}{1-x}$

with $x=2$, you substitute 2 for x:

$\frac{1-2^4}{1-2}$

Step-by-step method

Evaluate the exponent first:

$2^4 = 16$

So the numerator becomes:

$1-16=-15$

The denominator becomes:

$1-2=-1$

Now divide:

$\frac{-15}{-1}=15$

So the correct value is 15.

Useful algebra insight

This expression is a special rational expression. In fact,

$1-x^4=(1-x)(1+x+x^2+x^3)$

So when $x\ne 1$, the fraction simplifies to:

$\frac{1-x^4}{1-x}=1+x+x^2+x^3$

Then plugging in $x=2$ gives:

$1+2+4+8=15$

That is a faster way to verify your answer.

Common mistakes

Do not substitute 2 and then square the whole numerator or denominator. Also, do not forget that $1-2$ is negative. A double negative in the final division is what makes the answer positive.

Final check

Both direct substitution and factorization lead to the same result: 15.

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Pitfalls the pros know 👇 A frequent mistake is to compute the numerator and denominator in the wrong order or to miss the negative sign in $1-2$. Students also sometimes think the numerator should be $1-2^4= -15$ but then divide by 1 instead of by $-1$. Another subtle error is to treat the expression as if it were undefined because of the variable in the denominator. That is only a concern when the denominator becomes 0, which does not happen at $x=2$. Keep order of operations and sign handling separate, and the simplification stays manageable.

What if the problem changes? If the question changed to $\frac{1-x^4}{1-x}$ with $x=3$, then direct substitution gives $\frac{1-81}{1-3}=\frac{-80}{-2}=40$. If it changed to $\frac{1-x^5}{1-x}$, the expression would simplify to $1+x+x^2+x^3+x^4$ for $x\ne1$. These variants show that recognizing a factorization pattern can be faster than expanding everything from scratch.

Tags: polynomial evaluation, difference of powers, rational expression