91Ó°ÊÓ

In the context of factoring polynomials, a **common factor** is an aspect that can be extracted from each term within a polynomial. By identifying and factoring out the common factor, you simplify the expression. In our given exercise, the polynomial is \(2d^{4}+2d^{3}-60d^{2}\).
Every term in this polynomial shares a common factor of \(2d^{2}\).

• The first term is \(2d^{4}\), and can be divided by \(2d^2\) to yield \(d^2\).
• The second term is \(2d^{3}\), and reduces to \(d\) when you divide by \(2d^2\).
• The third term is \(-60d^{2}\), which simplifies to \(-30\) after dividing by \(2d^2\).

So, identifying the common factor is crucial as it enables the next stages of factoring the polynomial efficiently.

The next step in simplifying our polynomial involves a **quadratic equation**. After factoring out the common factor \(2d^{2}\), the expression simplifies from \(2d^{4}+2d^{3}-60d^{2}\) to \((2d^{2})(d^{2}+d-30)\). The quadratic part in question is \(d^{2}+d-30\).
Quadratic equations often need factoring themselves to simplify or solve further. The standard form of a quadratic equation is \(ax^{2}+bx+c\). Here, \(a\) is \(1\), \(b\) is \(1\), and \(c\) is \(-30\).
To factor \(d^{2}+d-30\), find two numbers that multiply to \(-30\) and add up to \(1\).

• These numbers are \(6\) and \(-5\).

Thus, the quadratic can be expressed as \((d+6)(d-5)\). Factoring quadratics like this helps break down the polynomial into its simplest form.

A **polynomial expression** is a mathematical phrase involving a sum of powers in one or more variables multiplied by coefficients. In the exercise \(2d^{4}+2d^{3}-60d^{2}\), each individual term is a part of the larger expression.
Polynomials can often look complex, but factoring is a technique that simplifies them. By identifying common factors and then breaking down components such as quadratics, polynomials become more manageable and can often be expressed in a more straightforward format.
To factor the given polynomial completely means to express it as the product of more straightforward expressions, such as in our exercise where it is factored into \((2d^{2})(d+6)(d-5)\).
The process involves:

• Finding the greatest common factor.
• Reducing the polynomial using that factor.
• Continuing to factor resulting parts such as quadratic expressions.

Understanding polynomials empowers you to tackle various algebraic challenges efficiently.