key term - X^2 + bx + c

Definition

The expression $x^2 + bx + c$ is a quadratic trinomial, where $x$ is the variable, and $b$ and $c$ are constants. This type of expression is the focus of the topic 7.2 Factor Trinomials of the Form $x^2 + bx + c$, which explores techniques for factoring such expressions into the product of two linear factors.

5 Must Know Facts For Your Next Test

1. The expression $x^2 + bx + c$ can be factored if the discriminant $b^2 - 4ac$ is a perfect square.
2. The factors of $x^2 + bx + c$ will be of the form $(x + m)(x + n)$, where $m$ and $n$ are the roots of the quadratic equation.
3. If the discriminant is negative, the expression $x^2 + bx + c$ cannot be factored over the real numbers and has complex conjugate roots.
4. Factoring $x^2 + bx + c$ is an important step in solving quadratic equations, as it allows for the use of the zero product property to find the solutions.
5. The process of factoring $x^2 + bx + c$ involves finding two numbers whose product is $ac$ and whose sum is $b$.

Review Questions

• Explain the relationship between the expression $x^2 + bx + c$ and the factorization of quadratic trinomials.
  • The expression $x^2 + bx + c$ is a quadratic trinomial, which can be factored into the product of two linear factors if the discriminant $b^2 - 4ac$ is a perfect square. The factors will be of the form $(x + m)(x + n)$, where $m$ and $n$ are the roots of the quadratic equation. Factoring $x^2 + bx + c$ is an important step in solving quadratic equations, as it allows for the use of the zero product property to find the solutions.
• Describe the role of the discriminant in the factorization of $x^2 + bx + c$.
  • The discriminant $b^2 - 4ac$ plays a crucial role in determining whether the expression $x^2 + bx + c$ can be factored over the real numbers. If the discriminant is a perfect square, the expression can be factored into the product of two linear factors. However, if the discriminant is negative, the expression cannot be factored over the real numbers and has complex conjugate roots. The nature of the roots, and consequently the factorization, is determined by the sign and value of the discriminant.
• Explain the process of factoring $x^2 + bx + c$ and how it relates to solving quadratic equations.
  • The process of factoring $x^2 + bx + c$ involves finding two numbers whose product is $ac$ and whose sum is $b$. These numbers will then be the coefficients of the linear factors, which will be of the form $(x + m)(x + n)$. Factoring $x^2 + bx + c$ is an important step in solving quadratic equations because it allows for the use of the zero product property to find the solutions. If the expression can be factored, the solutions to the quadratic equation will be the values of $x$ that make one of the factors equal to zero.

Related terms

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0.

Factoring:

The process of expressing a polynomial as the product of two or more simpler polynomials.

Discriminant:

The expression $b^2 - 4ac$ that determines the nature of the roots of a quadratic equation.