Factoring Trinomials with a ≠ 1

Factoring Trinomials with a ≠ 1 exercise

• Factor the quadratic function: $h(x) = 2x^2 + 4x - 6$.

  Hints

  Before applying the box method, replace $b$ with a sum of two factors of $ac$.

  Fill in a box with terms from the quadratic function and then find the greatest common factors of the columns and rows.

  Solution

  When presented with a quadratic function in standard form, the first step in factoring is to identify its coefficients; i.e. $a$, $b$, and $c$.

  We then need to find the product of $a$ and $c$, and then find the pair of factors of this product which sum to $b$.

  For example, with the equation $h(x) = 2x^2 + 4x - 6$, we have that $a = 2$, $b = 4$, and $c = -6$.

  Calculating the product of $a$ and $c$ gives us $ac = 2(-6) = -12$.

  So we need to find a pair of factors of $-12$ that sums to $4$.

  The pairs of factors of -12 are:

  $\begin{array}{rcrcr} 1 &\times & -12 &=& -12 \\ -1 & \times &12 &=& -12 \\ 2& \times & -6& =& -12 \\ -2& \times &6 &=& -12 \\ 3& \times & -4 &=& -12 \\ -3 & \times & 4 &=& -12 \end{array}$

  Summing each factor pair, we get:

  $\begin{array}{rcrcr} 1 & + & -12 &=& -11 \\ -1 &+& 12 &=& -11 \\ 2 &+& -6 &=& -4 \\ -2 &+& 6 &=& 4 \\ 3 &+& -4 &=& -1 \\ -3 &+& 4 &=& 1 \end{array}$

  The factor pair that sums to $b=4$ is $-2$ and $6$. With this pair, we can replace $b$ by their sum in our quadratic function:

  $h(x) = 2x^2 - 2x + 6x - 6$.

  Then we apply the box method to the trinomial, using the form of our function that has four terms to fill in the quadrants of the box.

  We find the greatest common factor (GCF) for each row and each column of the box, and sum the GCFs of the rows and the GCFs of the columns. The product of these sums gives us the factored form of our quadratic equation.

• Determine when a ball will land by using a quadratic function.

  Hints

  What is the height of the ball off the ground, if the ball is touching the ground?

  The right hand side of the equation is a product of two terms. When is the product of two terms equal to zero?

  Do you think that the flight of the ball is represented by the same function before the batter hits the ball?

  If the flight of the ball changes when the batter hits the ball, what does that mean about negative values of time in the equation? Keep in mind that time is equal to zero when the batter hits the ball, so a negative value of time occurs before the batter hits the ball.

  Solution

  Let's use the example equation given: $h(x) = (5x+1)(-x +3)$.

  When the ball hits the ground, its height above the ground will be equal to zero. We can represent this by saying that when the ball hits the ground,

  $h(x) = 0$.

  In order to find the time that the ball hits the ground, we need to find the value of $x$ that satisfies this equation.

  This will occur when $h(x)$ is equal to zero. Since the $h(x)$ is a product of two terms, it will be equal to zero when either (or both) of these terms is equal to zero.

  Therefore we can set each of these terms equal to zero and solve for $x$. With the first term, we have

  $5x + 1 = 0$.

  Subtracting $1$ from both sides, we get

  $5x = -1$.

  Dividing both sides by $5$, we get:

  $5x/5 = -1/5$,

  or

  $x = -1/5$.

  Similarly, with the second expression, we find that $x = 3$.

  We know that $h(x)$ is a function representing the height of the ball as a function of time. When the batter hits the ball, its flight changes completely. It doesn't make sense to have a negative value of time in our equation. Because before the batter hits the ball it will have a completely different trajectory that is not described by our function. So we must reject the value of $x$, or time, that is negative. This leaves us with the ball hitting the ground after three seconds.

  Whenever applying math to real-life situations it's important to think about your solutions and decide if they are reasonable.

• Use the box method to factor $h(x)=4x^2 + 7x - 2$.

  Hints

  Which blanks make the most sense to fill in first?

  What pair of factors do you need to find before you can begin to apply the box method?

  First find the pair of factors of $ac$ which sum to $b$. We have that

  $ac = 2\times 1 = 2$ and $b = 3$,

  so we need the pair of factors of $2$ that sum to $3$.

  The pairs of factors of $2$ are

  $1$ and $2$, and $-1$ and $-2$.

  Summing these pairs give us, $1 + 2 = 3$, and $-1 + (-2) = -3$.

  So the pair of factors of $ac$ that sum to $b$ is $1$ and $2$.

  Solution

  To apply the box method, we need to know all four terms that fill in the sections of the box. This means that we need to find a pair of factors of $ac$, which sum to $b$, so that we can replace the coefficient $b$ by their sum.

  For our particular function, $h(x)=4x^2 + 7x - 2$, we have that $ac = 4\times -2 = -8$.

  We must then find a pair of factors for this product that sums to $b$. Let's look at all factor pairs for this product:

  $\begin{array}{rcrcr} 2& \times &-4 &=& -8 \\ -2&\times &4 &= &-8 \\ 1&\times &-8 &= &-8 \\ -1&\times &8& = &-8 \end{array}$

  We need to find the pair of factors that sums to $b$. We can see that:

  $\begin{array}{rcrcr} 2 &+ &-4 &=& -2 \\ -2& + &4 &=& 2 \\ 1& - & 6 & =& -7 \\ -1&+&8& =& 7 \\ \end{array}$

  The term $b$ is equal to $7$, so the pair of factors of $ac$ which sum to $b$ is $8$ and $-1$.

  This means that our function becomes $h(x) = 4x^2 + 8x - x -2$ and the remaining two terms in our box become $8x$ and $-x$.

  We then need to find the greatest common factors of each row and column in our box. Let's put $8x$ in the upper right hand corner, and $-x$ in the lower left hand corner.

  The greatest common factor for the top row is $4x$ because this is the largest term that both $4x^2$ and $8x$ are divisible by. Likewise, the greatest common factor (GCF) for the bottom row is $-1$, the GCF for the first column is $x$, and the GCF for the second column is $2$.

  Now we find the product of the GCF column sum and the GCF row sum, which gives us the factored form of our function, $h(x) = (4x - 1)(x + 2)$.

• Factor the quadratic function $h(x) = -3x² + 4x + 4$.

  Hints

  Does the box method apply to this exercise?

  What pair of factors do you need to find before you can apply the box method?

  Solution

  We start with the equation $h(x) = -3x^2 + 4x + 4$.

  We need to find its factored form to help Samuel understand when the ball will hit the ground.

  The coefficients of $h(x)$'s standard form equation are $a = -3$, $b = 4$, and $c = 4$. The product of $a$ and $c$ is -12. So we know we need to find a pair of factors of $-12$ that sums to $4$. The pair of factors that satisfies these requirements is $6$ and $-2$.

  We then replace the coefficient $b$ from the standard form equation with the sum of $6$ and $-2$ and apply the box method using the terms of $h(x)=-3x^2 + 6x -2x + 4$.

  We must then find the greatest common factor for each row and column. The greatest common factor (GCF) for the first row is $3x$, because $3x$ is the largest term that both $-3x^2$ and $6x$ are divisible by. Similarly, we find the GCFs of each row and column.

  We then find the product of the GCF row sum and the GCF column sum. The product of these sums gives us the factored form of the quadratic equation:

  $h(x) =$(sum of the GCF of each row) $\times$ (sum of the GCF of each column)

  gives us

  $h(x) = (3x + 2)(-x + 2)$.

• Identify the standard form of a quadratic function.

  Hints

  Remember, the standard form has not been factored.

  The standard form is simplified as much as possible.

  Solution

  The function $h(x) = ax^2 + bx + c$ is in standard form.

  The functions $h(x) = (x-a) (x-b)$ and $h(x) = m (x-a) (x-b)$ are both in factored form. The factored form of a quadratic function provides the values of $x$ at which the function is equal to zero. These are called the roots, and are equal to $a$ and $b$. The second function is a special case of the third function, where $m = 1$.

  The function $h(x) = a(x-h)^2 + k$ is in vertex form.

• Find the factored form of each function written in standard form.

  Hints

  Pay close attention to the signs.

  Each standard form equation can be factored using the box method.

  Solution

  Let's use this function $h(x) = 3x^2 + 5x + 2$ as an example.

  First we need to find the pair of factors of $ac$ that sum to $b$. In the case of $h(x)$, we have $ac = 6$ and $b = 5$.

  The pairs of factors of $ac$ are:

  $\begin{array}{rcrcr} 1& \times &6 &=& 6\\ -1& \times &-6& =& 6\\ 2 &\times &3 &=& 6\\ -2& \times &-3& = &6 \end{array}$

  We need the pair of factors that sums to $5$. We can see that

  $\begin{array}{rcrcr} 1 &+ &6 &=& 7\\ -1 &+ &-6 &= &-7\\ 2 & + &3 &= &5\\ -2& + &-3 &= &-5 \end{array}$

  We than have that $2$ and $3$ is the pair of factors we need. We can now replace the coefficient $b$ with $3 + 2$, giving us the function $h(x) = 3x^2 + 3x +2x + 2$.

  Now that we have $h(x)$ written with four terms, we can factor using the box method. First, we fill in the four quadrants of the box, as in the image. Then we find the greatest common factor (GCF) of each row and column.

  The greatest common factor of the top row is $3x$, because this is the largest term that divides both $3x^2$ and $3x$. So we write $3x$ off to the left of the top row. Similarly, we find the GCFs of each row and column.

  Then we sum the GCFs of the rows and the GCFs of the columns. The product of these sums is our factored function.

  So, for our example, summing the row GCFs gives us

  $3x+2$,

  and summing the column GCFs gives us

  $x+1$.

  The product of these two sums gives us the factored form of our function:

  $h(x) = (3x+2)(x + 1)$.

  Similarly, you can find each remaining standard form and factored function pair:

  • $h(x) = 3x^2 + 5x + 2 = (3x+2)(x+1)$

  • $h(x) = 2x^2 -6x-8 = (2x+2)(x-4)$

  • $h(x) = 2x^2 -9x+9 = (2x-3)(x-3)$