91Ó°ÊÓ

When factoring algebraic expressions, finding a common factor is often the first step. A common factor is an expression that divides each term in the given polynomial without leaving a remainder. Identifying this factor simplifies the problem.

In the expression: \(y^{2}(y+1)-4 y(y+1)-21(y+1)\), each term shares a common factor of \((y+1)\). This common factor can be factored out as follows:
$$ y^{2}(y+1) - 4y(y+1) - 21(y+1) = (y+1)(y^{2} - 4y - 21) $$.

By factoring out \((y+1)\), the problem becomes simpler, focusing on the quadratic expression within parentheses.

A quadratic expression is a polynomial of degree 2, typically written in the form: $$ ax^2 + bx + c $$ where \(a\), \(b\), and \(c\) are constants. The primary goal when factoring a quadratic expression is to find two binomials that multiply to give the original expression.

For example, consider the quadratic expression from our problem: \( y^{2} - 4y - 21 \). We need to find two numbers that multiply to -21 (the constant term \(c\)) and add up to -4 (the coefficient \(b\)). These numbers are -7 and 3, because:

• \((-7) * 3 = -21\)
• \((-7) + 3 = -4\)

Hence, the quadratic expression can be factored as: $$ y^{2} - 4y - 21 = (y - 7)(y + 3) $$.

Factoring by grouping is a useful method when dealing with polynomials that cannot be easily factored through common techniques. This method involves grouping terms to find common factors within those groups.

In our problem, after factoring out the common factor \((y+1)\) from the original expression, we focus on the quadratic component: $$ y^{2} - 4y - 21 $$. This quadratic can be rewritten using the numbers we identified earlier -7 and 3.

We break it into groups as follows: $$ y^{2} - 4y - 21 $$ becomes $$ y^{2} - 7y + 3y - 21 $$ Next, we factor by grouping: $$ y(y - 7) + 3(y - 7) $$.

Notice that \((y - 7)\) is a common factor in each group, so we factor it out: $$ (y - 7)(y + 3) $$.

Combining all the factors gives us the final factored form of the original expression: $$ (y+1)(y-7)(y+3) $$. This systematic approach ensures we capture all components accurately, leading to the correct factorization.