Problem 16 - Algebra and Trigonometry

Question

Factor by grouping. $$x^{3}-x^{2}-5 x+5$$

Step-by-Step Solution

Answer

The factored form by grouping of the polynomial \( x^{3}-x^{2}-5 x+5 \) is \( (x^{2}-5)(x-1) \)

Step 1: Group the terms

Group the terms in the expression into two pairs. We get: \( (x^{3}-x^{2}) + (-5x+5) \) .

Step 2: Factor out the common terms

Factor out a common variable or number from each group. From the first group we can factor out \( x^{2} \). In the second group we can factor out \( -5\). This gives us: \( x^{2}(x-1) -5(x-1) \).

Step 3: Factor by grouping

Now, notice that the two chunks in our expression, \( x^{2} \) and \( -5 \), are both being multiplied by \( (x-1) \). Factor out \( (x-1) \) to get: \( (x^{2}-5)(x-1) \).