Lesson 8: Factoring Polynomials Completely

Property

HOW TO: Factor polynomials.

1. Step 1. Is there a greatest common factor? Factor it out.
2. Step 2. Is the polynomial a binomial, trinomial, or are there more than three terms? If it is a binomial, check for a sum/difference of squares or cubes. If it is a trinomial, check if it is of the form $x^2 + bx + c$ or $ax^2 + bx + c$. If it has more than three terms, use the grouping method.
3. Step 3. Check. Is it factored completely? Do the factors multiply back to the original polynomial?

Examples

• To factor $5x^6 + 15x^5$, first find the GCF, which is $5x^5$. Factoring it out gives $5x^5(x+3)$. The binomial $(x+3)$ is prime, so the factoring is complete.

• To factor $4x^2 - 8x - 60$, first factor out the GCF of 4, which gives $4(x^2 - 2x - 15)$. Then factor the trinomial to get $4(x-5)(x+3)$.

Property

To factor a trinomial of the form $x^2 + bx + c$ means to start with the product and end with the factors, $(x+m)(x+n)$. To get the correct factors, we find two numbers $m$ and $n$ whose product is $c$ and sum is $b$.

How to factor trinomials of the form $x^2 + bx + c$:

1. Write the factors as two binomials with first terms $x$: $(x \quad)(x \quad)$.
2. Find two numbers $m$ and $n$ that multiply to $c$ ($m \cdot n = c$) and add to $b$ ($m+n = b$).
3. Use $m$ and $n$ as the last terms of the factors: $(x+m)(x+n)$.
4. Check by multiplying the factors.

Examples

• To factor $x^2 + 9x + 20$, we need two numbers that multiply to 20 and add to 9. The numbers are 4 and 5. So, the factors are $(x+4)(x+5)$.
• To factor $a^2 - 8a + 15$, we need two numbers that multiply to 15 and add to -8. The numbers are -3 and -5. Thus, the factors are $(a-3)(a-5)$.
• To factor $p^2 + 3p - 10$, we need two numbers that multiply to -10 and add to 3. The numbers are 5 and -2. Therefore, the factors are $(p+5)(p-2)$.