The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
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The constant term is the term in a polynomial expression that does not contain any variables.
When multiplying or dividing polynomials, the constant term is multiplied or divided accordingly.
The constant term plays a crucial role in factoring trinomials, as it helps determine the factors of the expression.
In solving quadratic equations using the square root property or completing the square, the constant term is an essential component.
The constant term affects the y-intercept of the graph of a quadratic function, which is an important feature in graphing quadratic functions.
Review Questions
Explain the role of the constant term when multiplying polynomials.
When multiplying polynomials, the constant term is multiplied with the constant term of the other polynomial. This results in a new constant term in the product, which is the product of the two constant terms. For example, if multiplying $(2x^2 + 3x + 4)$ and $(x^2 - 2x + 5)$, the constant term in the product would be $4 \times 5 = 20$.
Describe how the constant term is used in the process of factoring trinomials.
In the process of factoring trinomials of the form $ax^2 + bx + c$, the constant term $c$ plays a crucial role. The factors of $c$ must be found such that their product is $a$ and their sum is $b$. This ensures that the trinomial can be expressed as the product of two binomials, which is the factored form of the expression.
Analyze the impact of the constant term on solving quadratic equations using the square root property or completing the square.
When solving quadratic equations of the form $ax^2 + bx + c = 0$, the constant term $c$ is essential. In the square root property method, the constant term is subtracted from both sides of the equation, creating a perfect square on the left side. In the completing the square method, the constant term is used to determine the value that must be added to both sides of the equation to create a perfect square. The constant term, therefore, directly influences the solutions obtained for the quadratic equation.