Problem 77 - Elementary Algebra

Question

Factor each trinomial of the form \(x^{2}+b x+c\). \(y^{2}-18 y+45\)

Step-by-Step Solution

Answer

The factored form is \((y-3)(y-15)\).

Step 1: Identify the coefficients

For the trinomial in the form \(x^{2}+bx+c\), recognize the coefficients in the given expression \(y^{2}-18y+45\). Here, \(b=-18\) and \(c=45\).

Step 2: Find two numbers that multiply to c and add to b

Find two numbers that multiply to \(45\) and add up to \(-18\). These numbers are \(-3\) and \(-15\) because \(-3 \times -15 = 45\) and \(-3 + -15 = -18\).

Step 3: Write the factored form using the numbers found

Using the numbers \(-3\) and \(-15\), rewrite the trinomial \(y^{2}-18y+45\) as \((y-3)(y-15)\).

Key Concepts

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves solving equations and understanding the relationships between variables. In algebra, variables like x and y are used to represent numbers in equations and expressions. This allows us to generalize mathematical concepts and solve a wide range of problems. For example, think about the equation \( y^2 - 18y + 45 \). Here, y is a variable, and the equation represents a quadratic polynomial.

quadratic expressions

Quadratic expressions are algebraic expressions of the form \( ax^2 + bx + c \), where a, b, and c are constants, and \( x \) is a variable. This type of expression is unique because it produces a parabola when graphed. The key feature of a quadratic expression is the \( x^2 \) term, which makes it a second-degree polynomial. For instance, in the trinomial \( y^2 - 18y + 45 \), the quadratic expression includes the \( y^2 \) term, making it a second-degree equation.

factoring techniques

Factoring techniques involve breaking down an expression into simpler components (factors) that, when multiplied together, yield the original expression. This method is crucial for solving quadratic equations. Step-by-step, factoring involves:

• Identifying the coefficients of the quadratic expression.
• Finding two numbers that multiply to the constant term but add to the linear coefficient.
• Rewriting the quadratic expression as a product of two binomials.

For instance, with \( y^2 - 18y + 45 \), we identify that \( b = -18 \) and \( c = 45 \). We then find two numbers that multiply to 45 and add to \( -18 \), which are \( -3 \) and \( -15 \). Finally, we rewrite the expression as \( (y-3)(y-15) \).

trinomial factoring

Trinomial factoring is a technique used to express a trinomial (a polynomial with three terms) as the product of two binomials. To factor a trinomial of the form \( x^2 + bx + c \), follow these steps:

• Identify the coefficients in the trinomial.
• Find two numbers that multiply to the constant term (c) and add to the linear coefficient (b).
• Use these two numbers to write the trinomial as a product of two binomials.

Looking at the specific example \( y^2 - 18y + 45 \), we identify \( b = -18 \) and \( c = 45 \). The numbers \( -3 \) and \( -15 \) fit the criteria, so we factor it as \( (y-3)(y-15) \). This process simplifies solving the equation and understanding its roots.