Solve 16x^6+24x^5+5x^4

$$16 x ^ { 6 } + 24 x ^ { 5 } + 5 x ^ { 4 }$$

Factor

$$\left(4x+1\right)\left(4x+5\right)x^{4}$$

Solution Steps

Factor out x^{4}.

$$x^{4}\left(16x^{2}+24x+5\right)$$

Consider 16x^{2}+24x+5. Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=24 ab=16\times 5=80$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 80.

$$1,80 2,40 4,20 5,16 8,10$$

Calculate the sum for each pair.

$$1+80=81 2+40=42 4+20=24 5+16=21 8+10=18$$

The solution is the pair that gives sum 24.

$$a=4 b=20$$

Rewrite 16x^{2}+24x+5 as \left(16x^{2}+4x\right)+\left(20x+5\right).

$$\left(16x^{2}+4x\right)+\left(20x+5\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(4x+1\right)+5\left(4x+1\right)$$

Factor out common term 4x+1 by using distributive property.

$$\left(4x+1\right)\left(4x+5\right)$$

Rewrite the complete factored expression.

$$x^{4}\left(4x+1\right)\left(4x+5\right)$$

Evaluate

$$\left(4x+1\right)\left(4x+5\right)x^{4}$$