Name: Division of a Trinomial by a Trinomial
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: Can we divide a polynomial by another polynomial to get the quotient polynomial and the remainder polynomial? Let's explore this concept in this video.

Question

Show that the quadratic polynomial

4 x^{2} + 12 x + 15

has no real zeroes.

Accepted Answer

We are given to show that the quadratic polynomial $4 x^{2} + 12 x + 15$ has no real zeroes.

For this, we shall draw the graph of the equation $y = 4 x^{2} + 12 x + 15$ . If the graph intersects or touches $x$ -axis, then there is a real solution. If the graph is completely above or below the $x$ -axis, the polynomial does not have a real solution.

For $x = 0, y = 4 {(0)}^{2} + 12 \times 0 + 15 = 15$

For $x = 1, y = 4 (1)^{2} + 12 \times 1 + 15 = 31$

For $x = - 1, y = 4 (- 1)^{2} + 12 \times (- 1) + 15 = 4 - 12 + 15 = 7$

For $x = 2, y = 4 (2)^{2} + 12 \times 2 + 15 = 16 + 24 + 15 = 55$

For $x = - 2, y = 4 (- 2)^{2} + 12 \times (- 2) + 15 = 16 - 24 + 15 = 7$

For $x = 3, y = 4 (3)^{2} + 12 \times 3 + 15 = 36 + 36 + 15 = 87$

For $x = - 3, y = 4 (- 3)^{2} + 12 \times (- 3) + 15 = 36 - 36 + 15 = 15$

For $x = 4, y = 4 (4)^{2} + 12 \times 4 + 15 = 64 + 48 + 15 = 127$

For $x = - 4, y = 4 (- 4)^{2} + 12 \times (- 4) + 15 = 64 - 48 + 15 = 31$

The equation can be written as follows:

$y = 4 x^{2} + 12 x + 15$

$\Rightarrow y = {(2 x + 3)}^{2} + 6$

$\Rightarrow y - 6 = (2 x + 3)^{2}$

$\Rightarrow$ The vertex of the graph is at $(- 1.5, 6)$ .

The table of values is given below:

$x$ $- 4$ $- 3$ $- 2$ $- 1$ $0$ $1$ $2$ $3$ $4$

$y$ $31$ $15$ $7$ $7$ $15$ $31$ $55$ $87$ $127$

Using all the points, we draw the graph of $y = 4 x^{2} + 12 x + 15$ as shown below:

We observe that the graph does not cut or intersect the $x$ -axis.

Therefore, the given quadratic polynomial has no real roots.

Show that the quadratic polynomial $4 x^{2} + 12 x + 15$ has no real zeroes.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Show that the quadratic polynomial 4x2+12x+15 has no real zeroes.

Important Questions on Polynomials

Learn and Prepare for any exam you want !

Show that the quadratic polynomial $4 x^{2} + 12 x + 15$ has no real zeroes.