Factor:
1) $x^2-11 x+28$
2) $9 x^3+21 x^2-60 x$
To factor the expression \(x^2 - 11x + 28\), we need to find two numbers whose product is \(28\) and whose sum is \(-11\). We look at the factors of \(28\):
Since the sum must be \(-11\) and the product must be positive, both factors must be negative.
Therefore, the quadratic expression factors into \((x - 4)(x - 7)\).
To factor the expression \(9x^3 + 21x^2 - 60x\), we first identify the greatest common factor (GCF).
Thus, the GCF is \(3x\). We factor this out from the expression:
\[9x^3 + 21x^2 - 60x = 3x(3x^2 + 7x - 20)\]
Next, we factor the quadratic \(3x^2 + 7x - 20\). We look for two numbers that multiply to \(3 \times (-20) = -60\) and add to \(7\).
The pairs of factors for -60 that satisfy the sum of 7 are -5 and 12.
We rewrite the middle term \(7x\) as \(-5x + 12x\):
\[3x^2 - 5x + 12x - 20\]
Now, factor by grouping:
\[x(3x - 5) + 4(3x - 5)\]
\[(x + 4)(3x - 5)\]
Combining these parts with our GCF, the complete factorization is \(3x(x + 4)(3x - 5)\).
Factoring is a fundamental tool in algebra used to simplify expressions and solve equations, which is helpful in various real-world scenarios. For example, if you are designing a rectangular garden and have an area defined by the expression \(x^2 - 11x + 28\), factoring it into \((x - 4)(x - 7)\) allows you to quickly identify possible dimensions, such as width \((x-4)\) and length \((x-7)\). Similarly, engineers often factor complex polynomial equations to find critical points, such as where an object's velocity or acceleration becomes zero, allowing for precise control and optimization in mechanical or structural systems.